Finite Group Databasebeta

← C2×C109×#1612 #42136113

Subgroups — C2×C109×#1612 #42136113

Order 27904.

1156 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#538×C109C211yesmaximal, characteristic
2C2×#483×C109C211yesmaximal, characteristic, complemented
3C2×#480×C109C211yesmaximal, characteristic, complemented
4C2×#397×C109C211yesmaximal, characteristic, complemented
5C109×#1612C218yesmaximal, direct product factor
6C2×#328×C109C211yesmaximal, characteristic, complemented
7C2×#327×C109C211yesmaximal, characteristic, complemented
8C2×#323×C109C211yesmaximal, characteristic, complemented
9#538×C109—22no—
10#538×C109—22no—
11#538×C109—22no—
12#538×C109C2214yes—
13C2×#126×C109—21no—
14C2×#124×C109—21no—
15C2×#123×C109—21no—
16C2×C22⋊Q8×C109C2211yescharacteristic
17C2×#121×C109—21no—
18C2×#121×C109C2211yescharacteristic
19C2×#120×C109—21no—
20C2×C4×D4×C109—21no—
21C22×C4⋊C4×C109C2211yescharacteristic
22C22×C22⋊C4×C109—21no—
23C22×C22⋊C4×C109C2211yescharacteristic
24#483×C109—22no—
25#483×C109C2214yescomplemented
26#480×C109—22no—
27#480×C109C2214yescomplemented
28C2×#107×C109—21no—
29C2×#106×C109—21no—
30C2×#103×C109—21no—
31C2×#102×C109—21no—
32C2×C22⋊C8×C109C2211yescharacteristic
33C2×C22⋊C8×C109C2211yescharacteristic
34#397×C109—22no—
35#397×C109C2214yescomplemented
36C2×#95×C109—21no—
37C2×#95×C109C2211yescharacteristic
38#328×C109C2214yescomplemented
39#327×C109C2214yescomplemented
40#323×C109C2214yescomplemented
41C25×C109D411yescharacteristic
42C22×Q8×C109D411yescharacteristic
43C22×D4×C109—21no—
44C22×D4×C109—21no—
45C22×D4×C109—21no—
46C22×D4×C109D411yescharacteristic
47C23×C4×C109—21no—
48C23×C4×C109—21no—
49C23×C4×C109—21no—
50C23×C4×C109D411yescharacteristic
51C23×C4×C109C2311yescharacteristic
52C22×C8×C109—21no—
53C22×C8×C109—21no—
54#126×C109—24no—
55#126×C109—24no—
56#124×C109—24no—
57#124×C109—24no—
58#123×C109—24no—
59#123×C109—24no—
60C22⋊Q8×C109—21no—
61C22⋊Q8×C109—21no—
62C22⋊Q8×C109D412yes—
63C22⋊Q8×C109C2312yes—
64#121×C109—24no—
65#121×C109—24no—
66#121×C109—21no—
67#121×C109—21no—
68#121×C109D412yes—
69#121×C109C2312yes—
70#120×C109—24no—
71#120×C109—24no—
72C4×D4×C109—24no—
73C4×D4×C109—24no—
74C2×C4⋊C4×C109—21no—
75C2×C4⋊C4×C109—21no—
76C2×C4⋊C4×C109—21no—
77C2×C4⋊C4×C109—21no—
78C2×C4⋊C4×C109—21no—
79C2×C4⋊C4×C109—21no—
80C2×C4⋊C4×C109—21no—
81C2×C4⋊C4×C109D411yescharacteristic
82C2×C4⋊C4×C109D411yescharacteristic
83C2×C4⋊C4×C109C2312yes—
84C2×C22⋊C4×C109—21no—
85C2×C22⋊C4×C109—21no—
86C2×C22⋊C4×C109—21no—
87C2×C22⋊C4×C109—21no—
88C2×C22⋊C4×C109—21no—
89C2×C22⋊C4×C109—21no—
90C2×C22⋊C4×C109—21no—
91C2×C22⋊C4×C109—21no—
92C2×C22⋊C4×C109—21no—
93C2×C22⋊C4×C109—24no—
94C2×C22⋊C4×C109—44no—
95C2×C22⋊C4×C109D412yes—
96C2×C22⋊C4×C109C2312yes—
97C2×C42×C109—21no—
98#107×C109—24no—
99#106×C109—24no—
100#103×C109—24no—
101#102×C109—24no—
102C22⋊C8×C109—21no—
103C22⋊C8×C109—21no—
104C22⋊C8×C109C2312yes—
105C22⋊C8×C109C2312yes—
106#95×C109—24no—
107#95×C109—21no—
108#95×C109C2312yes—
109C24×C109—21no—
110C24×C109—21no—
111C24×C109—22no—
112C24×C109—21no—
113C24×C109—22no—
114C24×C109—42no—
115C24×C109—42no—
116C24×C109C2×D411yescharacteristic
117C24×C109C2×D412yes—
118C2×Q8×C109—21no—
119C2×Q8×C109—42no—
120C2×Q8×C109C2×D412yes—
121C2×D4×C109—22no—
122C2×D4×C109—22no—
123C2×D4×C109—24no—
124C2×D4×C109—24no—
125C2×D4×C109—21no—
126C2×D4×C109—42no—
127C2×D4×C109—42no—
128C2×D4×C109—42no—
129C2×D4×C109—82no—
130C2×D4×C109—82no—
131C2×D4×C109C2×D412yes—
132C22×C4×C109—21no—
133C22×C4×C109—21no—
134C22×C4×C109—21no—
135C22×C4×C109—21no—
136C22×C4×C109—24no—
137C22×C4×C109—21no—
138C22×C4×C109—22no—
139C22×C4×C109—22no—
140C22×C4×C109—42no—
141C22×C4×C109—42no—
142C22×C4×C109—42no—
143C22×C4×C109—42no—
144C22×C4×C109—41no—
145C22×C4×C109—41no—
146C22×C4×C109—42no—
147C22×C4×C109—42no—
148C22×C4×C109—42no—
149C22×C4×C109—82no—
150C22×C4×C109C2×D411yescharacteristic
151C22×C4×C109C2×D411yescharacteristic
152C22×C4×C109C2×D412yes—
153C22×C4×C109C2×D411yescharacteristic
154C22×C4×C109C2×D411yescharacteristic
155C22×C4×C109C2×D411yescharacteristic
156C22×C4×C109C2411yescharacteristic
157C2×C8×C109—22no—
158C2×C8×C109—22no—
159C2×C8×C109—42no—
160C2×C8×C109—42no—
161C4⋊C4×C109—24no—
162C4⋊C4×C109—22no—
163C4⋊C4×C109—21no—
164C4⋊C4×C109—21no—
165C4⋊C4×C109—22no—
166C4⋊C4×C109—22no—
167C4⋊C4×C109—22no—
168C4⋊C4×C109—41no—
169C4⋊C4×C109—41no—
170C4⋊C4×C109C2×D412yes—
171C4⋊C4×C109C2×D412yes—
172C22⋊C4×C109—24no—
173C22⋊C4×C109—24no—
174C22⋊C4×C109—24no—
175C22⋊C4×C109—22no—
176C22⋊C4×C109—22no—
177C22⋊C4×C109—22no—
178C22⋊C4×C109—22no—
179C22⋊C4×C109—22no—
180C22⋊C4×C109—22no—
181C22⋊C4×C109—22no—
182C22⋊C4×C109—22no—
183C22⋊C4×C109—48no—
184C42×C109—24no—
185C23×C109—22no—
186C23×C109—22no—
187C23×C109—22no—
188C23×C109—22no—
189C23×C109—22no—
190C23×C109—22no—
191C23×C109—42no—
192C23×C109—42no—
193C23×C109—48no—
194C23×C109—42no—
195C23×C109—42no—
196C23×C109—42no—
197C23×C109—42no—
198C23×C109—42no—
199C23×C109—42no—
200C23×C109—42no—
201C23×C109—81no—
202C23×C109—81no—
203C23×C109—84no—
204C23×C109#12011yescharacteristic
205C23×C109#12011yescharacteristic
206C23×C109#13512yes—
207C23×C109AGL(1,ℤ/8)12yes—
208C23×C109C22×D411yescharacteristic
209Q8×C109—42no—
210Q8×C109—81no—
211D4×C109—42no—
212D4×C109—42no—
213D4×C109—42no—
214D4×C109—44no—
215D4×C109—81no—
216D4×C109—84no—
217D4×C109—84no—
218C2×C4×C109—22no—
219C2×C4×C109—22no—
220C2×C4×C109—22no—
221C2×C4×C109—22no—
222C2×C4×C109—22no—
223C2×C4×C109—22no—
224C2×C4×C109—41no—
225C2×C4×C109—41no—
226C2×C4×C109—41no—
227C2×C4×C109—41no—
228C2×C4×C109—42no—
229C2×C4×C109—42no—
230C2×C4×C109—42no—
231C2×C4×C109—42no—
232C2×C4×C109—42no—
233C2×C4×C109—42no—
234C2×C4×C109—42no—
235C2×C4×C109—42no—
236C2×C4×C109—42no—
237C2×C4×C109—42no—
238C2×C4×C109—42no—
239C2×C4×C109—81no—
240C2×C4×C109—84no—
241C2×C4×C109—84no—
242C2×C4×C109—82no—
243C2×C4×C109—82no—
244C2×C4×C109#12011yescharacteristic
245C2×C4×C109#12011yescharacteristic
246C2×C4×C109C22×D411yescharacteristic
247C2×C4×C109C22×D411yescharacteristic
248C8×C109—42no—
249C8×C109—42no—
250C22×C109—24no—
251C22×C109—24no—
252C22×C109—21no—
253C22×C109—21no—
254C22×C109—21no—
255C22×C109—21no—
256C22×C109—44no—
257C22×C109—42no—
258C22×C109—42no—
259C22×C109—42no—
260C22×C109—42no—
261C22×C109—42no—
262C22×C109—42no—
263C22×C109—82no—
264C22×C109—84no—
265C22×C109—84no—
266C22×C109—84no—
267C22×C109#44912yes—
268C22×C109#44912yes—
269C22×C109#45512yes—
270C22×C109#45812yes—
271C22×C109C2×#12011yescharacteristic
272C22×C109C2×#13511yescharacteristic
273C22×C109C2×AGL(1,ℤ/8)11yescharacteristic
274C4×C109—21no—
275C4×C109—21no—
276C4×C109—21no—
277C4×C109—21no—
278C4×C109—42no—
279C4×C109—42no—
280C4×C109—42no—
281C4×C109—42no—
282C4×C109—82no—
283C4×C109—82no—
284C2×#1612C10911yesmaximal, characteristic, direct product factor, Sylow
285C2×C109—21no—
286C2×C109—21no—
287C2×C109—22no—
288C2×C109—82no—
289C2×C109—82no—
290C2×C109#161214yesdirect product factor
291C2×C109C2×#44911yescharacteristic
292C2×C109C2×#45511yescharacteristic
293C2×C109C2×#45811yescharacteristic
294C2×#538C2×C10911yescharacteristic
295C2×#483C2×C10911yescharacteristic, complemented
296C2×#480C2×C10911yescharacteristic, complemented
297C2×#397C2×C10911yescharacteristic, complemented
298#1612C2×C10918yesdirect product factor
299C2×#328C2×C10911yescharacteristic, complemented
300C2×#327C2×C10911yescharacteristic, complemented
301C2×#323C2×C10911yescharacteristic, complemented
302C109C2×#161211yescharacteristic, direct product factor, Sylow
303#538—22no—
304#538—22no—
305#538—22no—
306#538C22×C10914yes—
307C2×#126—21no—
308C2×#124—21no—
309C2×#123—21no—
310C2×C22⋊Q8C22×C10911yescharacteristic
311C2×#121—21no—
312C2×#121C22×C10911yescharacteristic
313C2×#120—21no—
314C2×C4×D4—21no—
315C22×C4⋊C4C22×C10911yescharacteristic
316C22×C22⋊C4—21no—
317C22×C22⋊C4C22×C10911yescharacteristic
318#483—22no—
319#483C22×C10914yescomplemented
320#480—22no—
321#480C22×C10914yescomplemented
322C2×#107—21no—
323C2×#106—21no—
324C2×#103—21no—
325C2×#102—21no—
326C2×C22⋊C8C22×C10911yescharacteristic
327C2×C22⋊C8C22×C10911yescharacteristic
328#397—22no—
329#397C22×C10914yescomplemented
330C2×#95—21no—
331C2×#95C22×C10911yescharacteristic
332#328C22×C10914yescomplemented
333#327C22×C10914yescomplemented
334#323C22×C10914yescomplemented
335C25D4×C10911yescharacteristic
336C22×Q8D4×C10911yescharacteristic
337C22×D4—21no—
338C22×D4—21no—
339C22×D4—21no—
340C22×D4D4×C10911yescharacteristic
341C23×C4—21no—
342C23×C4—21no—
343C23×C4—21no—
344C23×C4D4×C10911yescharacteristic
345C23×C4C23×C10911yescharacteristic
346C22×C8—21no—
347C22×C8—21no—
348#126—24no—
349#126—24no—
350#124—24no—
351#124—24no—
352#123—24no—
353#123—24no—
354C22⋊Q8—21no—
355C22⋊Q8—21no—
356C22⋊Q8D4×C10912yes—
357C22⋊Q8C23×C10912yes—
358#121—24no—
359#121—24no—
360#121—21no—
361#121—21no—
362#121D4×C10912yes—
363#121C23×C10912yes—
364#120—24no—
365#120—24no—
366C4×D4—24no—
367C4×D4—24no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4D4×C10911yescharacteristic
376C2×C4⋊C4D4×C10911yescharacteristic
377C2×C4⋊C4C23×C10912yes—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—24no—
388C2×C22⋊C4—44no—
389C2×C22⋊C4D4×C10912yes—
390C2×C22⋊C4C23×C10912yes—
391C2×C42—21no—
392#107—24no—
393#106—24no—
394#103—24no—
395#102—24no—
396C22⋊C8—21no—
397C22⋊C8—21no—
398C22⋊C8C23×C10912yes—
399C22⋊C8C23×C10912yes—
400#95—24no—
401#95—21no—
402#95C23×C10912yes—
403C24—21no—
404C24—21no—
405C24—22no—
406C24—21no—
407C24—22no—
408C24—42no—
409C24—42no—
410C24C2×D4×C10911yescharacteristic
411C24C2×D4×C10912yes—
412C2×Q8—21no—
413C2×Q8—42no—
414C2×Q8C2×D4×C10912yes—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—24no—
418C2×D4—24no—
419C2×D4—21no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—82no—
424C2×D4—82no—
425C2×D4C2×D4×C10912yes—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—24no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—82no—
444C22×C4C2×D4×C10911yescharacteristic
445C22×C4C2×D4×C10911yescharacteristic
446C22×C4C2×D4×C10912yes—
447C22×C4C2×D4×C10911yescharacteristic
448C22×C4C2×D4×C10911yescharacteristic
449C22×C4C2×D4×C10911yescharacteristic
450C22×C4C24×C10911yescharacteristic
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—42no—
454C2×C8—42no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4C2×D4×C10912yes—
465C4⋊C4C2×D4×C10912yes—
466C22⋊C4—24no—
467C22⋊C4—24no—
468C22⋊C4—24no—
469C22⋊C4—22no—
470C22⋊C4—22no—
471C22⋊C4—22no—
472C22⋊C4—22no—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—48no—
478C42—24no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—42no—
486C23—42no—
487C23—48no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—81no—
496C23—81no—
497C23—84no—
498C23#120×C10911yescharacteristic
499C23#120×C10911yescharacteristic
500C23#135×C10912yes—
501C23AGL(1,ℤ/8)×C10912yes—
502C23C22×D4×C10911yescharacteristic
503Q8—42no—
504Q8—81no—
505D4—42no—
506D4—42no—
507D4—42no—
508D4—44no—
509D4—81no—
510D4—84no—
511D4—84no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—81no—
534C2×C4—84no—
535C2×C4—84no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4#120×C10911yescharacteristic
539C2×C4#120×C10911yescharacteristic
540C2×C4C22×D4×C10911yescharacteristic
541C2×C4C22×D4×C10911yescharacteristic
542C8—42no—
543C8—42no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—24no—
549C22—24no—
550C22—44no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—82no—
558C22—84no—
559C22—84no—
560C22—84no—
561C22#449×C10912yes—
562C22#449×C10912yes—
563C22#455×C10912yes—
564C22#458×C10912yes—
565C22C2×#120×C10911yescharacteristic
566C22C2×#135×C10911yescharacteristic
567C22C2×AGL(1,ℤ/8)×C10911yescharacteristic
568C4—21no—
569C4—21no—
570C4—21no—
571C4—21no—
572C4—42no—
573C4—42no—
574C4—42no—
575C4—42no—
576C4—82no—
577C4—82no—
578C2—21no—
579C2—21no—
580C2—22no—
581C2—82no—
582C2—82no—
583C2C109×#161214yesdirect product factor
584C2C2×#449×C10911yescharacteristic
585C2C2×#455×C10911yescharacteristic
586C2C2×#458×C10911yescharacteristic
subgroupG/H
center, socleC23×C109#120
commutator, Frattini, cosocleC22×C4C24×C109
prime radicalC2×#538×C109C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C109 ◁ C23×C4×C109 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#483×C109C2semidirect
3. C2×#480×C109C2semidirect
4. C2×#397×C109C2semidirect
5. C109×#1612C2direct (N × H)
6. C2×#328×C109C2semidirect
7. C2×#327×C109C2semidirect
8. C2×#323×C109C2semidirect
25. #483×C109C22semidirect
27. #480×C109C22semidirect
35. #397×C109C22semidirect
38. #328×C109C22semidirect
39. #327×C109C22semidirect
40. #323×C109C22semidirect
284. C2×#1612C109direct (N × H)
290. C2×C109#1612direct (N × H)
295. C2×#483C2×C109semidirect
296. C2×#480C2×C109semidirect
297. C2×#397C2×C109semidirect
298. #1612C2×C109direct (N × H)
299. C2×#328C2×C109semidirect
300. C2×#327C2×C109semidirect
301. C2×#323C2×C109semidirect
302. C109C2×#1612direct (N × H)
319. #483C22×C109semidirect
321. #480C22×C109semidirect
329. #397C22×C109semidirect
332. #328C22×C109semidirect
333. #327C22×C109semidirect
334. #323C22×C109semidirect
583. C2C109×#1612direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#538×C1092
C2×#483×C1092
C2×#480×C1092
C2×#397×C1092
C109×#16122
C2×#328×C1092
C2×#327×C1092
C2×#323×C1092
C2×#1612109