Finite Group Databasebeta

← #46839799

Subgroups — #46839799

Order 34688.

788 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576135C211yesmaximal, characteristic, complemented
2C2×#22576133C211yesmaximal, characteristic, complemented
3C2×#22576133C211yesmaximal, characteristic, complemented
4C2×#22576133C211yesmaximal, characteristic, complemented
5C2×#22576122C211yesmaximal, characteristic, complemented
6C2×C4⋊C4×D271C211yesmaximal, characteristic, complemented
7#31773819C211yesmaximal, characteristic, complemented
8#31773814C211yesmaximal, characteristic, complemented
9#31773813C211yesmaximal, characteristic, complemented
10#31773797C211yesmaximal, characteristic, complemented
11#31773792C211yesmaximal, characteristic, complemented
12#31773775C211yesmaximal, characteristic, complemented
13#31773773C211yesmaximal, characteristic, complemented
14#31773756C211yesmaximal, characteristic, complemented
15#396×C271C211yesmaximal, characteristic, complemented
16C22×D1084C2211yescharacteristic
17C22×D1084C2211yescharacteristic
18C22×D1084C2211yescharacteristic
19C22×C22⋊D271C2211yescharacteristic
20C22×C22⋊D271C2211yescharacteristic
21C22×C22⋊D271C2211yescharacteristic
22C22×C4×D271C2211yescharacteristic
23C22×C4×D271C2211yescharacteristic
24C22×C4×D271C2211yescharacteristic
25C22×C4×D271C2211yescharacteristic
26#22576135—24no—
27#22576133—24no—
28#22576133—24no—
29#22576133—24no—
30#22576122—24no—
31C2×C4⋊Dic271C2211yescharacteristic
32C2×Dic271⋊C4C2211yescharacteristic
33C2×Dic271⋊C4C2211yescharacteristic
34C2×Dic271⋊C4C2211yescharacteristic
35C2×Dic271⋊C4C2211yescharacteristic
36C4⋊C4×D271—24no—
37C2×C22⋊Dic271C2211yescharacteristic
38C2×(C2×D271)⋊C4C2211yescharacteristic
39C2×(C2×D271)⋊C4C2211yescharacteristic
40C2×(C2×D271)⋊C4C2211yescharacteristic
41C2×(C2×D271)⋊C4C2211yescharacteristic
42C2×(C2×D271)⋊C4C2211yescharacteristic
43C2×(C2×D271)⋊C4C2211yescharacteristic
44C2×(C2×D271)⋊C4C2211yescharacteristic
45C2×(C2×D271)⋊C4C2211yescharacteristic
46C2×C4×Dic271C2211yescharacteristic
47C271⋊#95C2211yescharacteristic
48C271⋊#95C2211yescharacteristic
49C271⋊#95C2211yescharacteristic
50C2×C4⋊C4×C271C2211yescharacteristic
51C2×C4⋊C4×C271C2211yescharacteristic
52C2×C4⋊C4×C271C2211yescharacteristic
53C2×C22⋊C4×C271C2211yescharacteristic
54C2×C22⋊C4×C271C2211yescharacteristic
55C2×C22⋊C4×C271C2211yescharacteristic
56#95×C271C2211yescharacteristic
57C23×D271C2311yescharacteristic
58C23×D271C2311yescharacteristic
59C23×D271C2311yescharacteristic
60C2×D1084—22no—
61C2×D1084—22no—
62C2×D1084—22no—
63C2×D1084—22no—
64C2×D1084—22no—
65C2×D1084—22no—
66C2×D1084—22no—
67C2×D1084—41no—
68C2×D1084D414yescomplemented
69C2×C22⋊D271—22no—
70C2×C22⋊D271—22no—
71C2×C22⋊D271—22no—
72C2×C22⋊D271—22no—
73C2×C22⋊D271—22no—
74C2×C22⋊D271—22no—
75C2×C22⋊D271—22no—
76C2×C22⋊D271—22no—
77C2×C22⋊D271—41no—
78C22×Dic271C2311yescharacteristic
79C22×Dic271C2311yescharacteristic
80C22×Dic271C2311yescharacteristic
81C22×Dic271C2311yescharacteristic
82C22×Dic271C2311yescharacteristic
83C2×C4×D271—22no—
84C2×C4×D271—22no—
85C2×C4×D271—22no—
86C2×C4×D271—22no—
87C2×C4×D271—22no—
88C2×C4×D271—22no—
89C2×C4×D271—22no—
90C2×C4×D271—22no—
91C2×C4×D271—22no—
92C2×C4×D271—41no—
93C2×C4×D271—41no—
94C2×C4×D271D414yescomplemented
95C4⋊Dic271—22no—
96Dic271⋊C4—22no—
97Dic271⋊C4—22no—
98Dic271⋊C4—22no—
99Dic271⋊C4D414yescomplemented
100C22⋊Dic271—41no—
101(C2×D271)⋊C4—22no—
102(C2×D271)⋊C4—22no—
103(C2×D271)⋊C4—22no—
104(C2×D271)⋊C4—22no—
105(C2×D271)⋊C4—22no—
106(C2×D271)⋊C4—22no—
107(C2×D271)⋊C4—22no—
108(C2×D271)⋊C4—41no—
109C4×Dic271—22no—
110C24×C271C2311yescharacteristic
111C22×C4×C271C2311yescharacteristic
112C22×C4×C271C2311yescharacteristic
113C22×C4×C271C2311yescharacteristic
114C22×C4×C271C2311yescharacteristic
115C22×C4×C271C2311yescharacteristic
116C22×C4×C271C2311yescharacteristic
117C4⋊C4×C271—22no—
118C4⋊C4×C271—22no—
119C4⋊C4×C271—22no—
120C22⋊C4×C271—22no—
121C22⋊C4×C271—22no—
122C22⋊C4×C271—22no—
123C22×D271—21no—
124C22×D271—21no—
125C22×D271—21no—
126C22×D271—21no—
127C22×D271—21no—
128C22×D271—21no—
129C22×D271—21no—
130C22×D271—21no—
131C22×D271—21no—
132C22×D271—21no—
133C22×D271—21no—
134C22×D271—21no—
135C22×D271—21no—
136C22×D271—21no—
137C22×D271—21no—
138C22×D271—21no—
139C22×D271—21no—
140C22×D271C2×D412yes—
141C22×D271C2×D412yes—
142C22×D271C2×D412yes—
143C22×D271#4112yes—
144D1084—28no—
145D1084—44no—
146D1084—44no—
147C22⋊D271—44no—
148C22⋊D271—44no—
149C22⋊D271—44no—
150C2×Dic271—21no—
151C2×Dic271—21no—
152C2×Dic271—21no—
153C2×Dic271—21no—
154C2×Dic271—21no—
155C2×Dic271—21no—
156C2×Dic271—21no—
157C2×Dic271—21no—
158C2×Dic271—21no—
159C2×Dic271—21no—
160C2×Dic271—21no—
161C2×Dic271C2×D412yes—
162C2×Dic271C2×D412yes—
163C2×Dic271C2×D412yes—
164C2×Dic271#4112yes—
165C4×D271—28no—
166C4×D271—44no—
167C4×D271—44no—
168C4×D271—44no—
169C23×C271—21no—
170C23×C271—21no—
171C23×C271—21no—
172C23×C271—21no—
173C23×C271—21no—
174C23×C271—21no—
175C23×C271—21no—
176C23×C271C2411yescharacteristic
177C2×C4×C271—21no—
178C2×C4×C271—21no—
179C2×C4×C271—21no—
180C2×C4×C271—21no—
181C2×C4×C271—21no—
182C2×C4×C271—21no—
183C2×C4×C271—21no—
184C2×C4×C271—21no—
185C2×C4×C271—21no—
186C2×C4×C271—21no—
187C2×C4×C271—21no—
188C2×C4×C271—21no—
189C2×C4×C271—21no—
190C2×C4×C271—21no—
191C2×C4×C271—21no—
192C2×C4×C271C2×D412yes—
193C2×C4×C271C2×D412yes—
194C2×C4×C271C2×D412yes—
195C22×C271—41no—
196C22×C271—41no—
197C22×C271—41no—
198C22×C271—41no—
199C22×C271—41no—
200C22×C271—41no—
201C22×C271—41no—
202C22×C271C22×D411yescharacteristic
203C22×C271C22×D411yescharacteristic
204C22×C271C22×D411yescharacteristic
205C22×C271C2×#4111yescharacteristic
206C22×C271C2×#4111yescharacteristic
207C22×C271H(5,𝔽2)11yescharacteristic
208C22×C2712−1+411yescharacteristic
209C2×D271—22no—
210C2×D271—22no—
211C2×D271—22no—
212C2×D271—22no—
213C2×D271—22no—
214C2×D271—22no—
215C2×D271—22no—
216C2×D271—22no—
217C2×D271—22no—
218C2×D271—41no—
219C2×D271—41no—
220C2×D271—41no—
221C2×D271—41no—
222C2×D271—41no—
223C2×D271—41no—
224C2×D271—41no—
225C2×D271—41no—
226C2×D271—41no—
227C2×D271—41no—
228C2×D271—41no—
229C2×D271#12114yescomplemented
230C4×C271—22no—
231C4×C271—22no—
232C4×C271—22no—
233C4×C271—41no—
234C4×C271—41no—
235C4×C271—41no—
236Dic271—22no—
237Dic271—41no—
238Dic271—41no—
239Dic271—41no—
240Dic271#12114yescomplemented
241C2×C271—81no—
242C2×C271C2×#12111yescharacteristic
243C2×C271C2×#12111yescharacteristic
244C2×C271#53711yescharacteristic
245C2×C271D4211yescharacteristic
246C2×C271#54711yescharacteristic
247C2×C271#54811yescharacteristic
248C2×C271#55311yescharacteristic
249D271—24no—
250D271—42no—
251D271—81no—
252C271#241611yescharacteristic, complemented, Sylow
253#2416—2711nomaximal, Sylow
254C2×#121—2711no—
255C2×#121—2711no—
256C2×#121—2711no—
257C2×#121—2711no—
258C2×C4×D4—2711no—
259C22×C4⋊C4—2711no—
260#396—2711no—
261#396D27111yescharacteristic, complemented
262#394—2711no—
263#394—2711no—
264#390—2711no—
265#389—2711no—
266#386—2711no—
267#386—2711no—
268#380—2711no—
269C22×D4—2711no—
270C22×D4—2711no—
271C22×D4—2711no—
272C22×D4—2711no—
273C22×D4—2711no—
274C22×D4—2711no—
275C23×C4—2711no—
276C23×C4—2711no—
277C23×C4—2711no—
278C23×C4—2711no—
279#121—5424no—
280#121—5424no—
281#121—5424no—
282#121—5424no—
283C4×D4—5424no—
284C2×C4⋊C4—2711no—
285C2×C4⋊C4—2711no—
286C2×C4⋊C4—2711no—
287C2×C4⋊C4—2711no—
288C2×C4⋊C4—2711no—
289C2×C4⋊C4—5424no—
290C2×C4⋊C4C2×D27111yescharacteristic
291C2×C4⋊C4C2×D27111yescharacteristic
292C2×C4⋊C4C2×D27111yescharacteristic
293C2×C22⋊C4—2711no—
294C2×C22⋊C4—2711no—
295C2×C22⋊C4—2711no—
296C2×C22⋊C4—2711no—
297C2×C22⋊C4—2711no—
298C2×C22⋊C4—2711no—
299C2×C22⋊C4—2711no—
300C2×C22⋊C4—2711no—
301C2×C22⋊C4—2711no—
302C2×C22⋊C4C2×D27111yescharacteristic
303C2×C22⋊C4C2×D27111yescharacteristic
304C2×C22⋊C4C2×D27111yescharacteristic
305C2×C42—2711no—
306#95—2711no—
307#95—2711no—
308#95—2711no—
309#95C2×D27111yescharacteristic
310C24—2711no—
311C24—2711no—
312C24—2711no—
313C24C22×D27111yescharacteristic
314C2×D4—2714no—
315C2×D4—5422no—
316C2×D4—5422no—
317C2×D4—5422no—
318C2×D4—5422no—
319C2×D4—5422no—
320C2×D4—5422no—
321C2×D4—5422no—
322C2×D4—5422no—
323C2×D4—5422no—
324C2×D4—5422no—
325C2×D4—5422no—
326C2×D4—5422no—
327C2×D4—5422no—
328C2×D4—5422no—
329C2×D4—5422no—
330C2×D4—10841no—
331C2×D4—10841no—
332C22×C4—2711no—
333C22×C4—2711no—
334C22×C4—2714no—
335C22×C4—2711no—
336C22×C4—2711no—
337C22×C4—2711no—
338C22×C4—5422no—
339C22×C4—5422no—
340C22×C4—5422no—
341C22×C4—5422no—
342C22×C4—5422no—
343C22×C4—5422no—
344C22×C4—5422no—
345C22×C4—5422no—
346C22×C4—5422no—
347C22×C4—10841no—
348C22×C4—10841no—
349C22×C4C22×D27111yescharacteristic
350C22×C4C22×D27111yescharacteristic
351C22×C4C22×D27111yescharacteristic
352C22×C4C22×D27111yescharacteristic
353C22×C4C22×D27111yescharacteristic
354C22×C4C22×D27111yescharacteristic
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—2714no—
359C4⋊C4—5422no—
360C4⋊C4—5422no—
361C4⋊C4—5422no—
362C4⋊C4—5422no—
363C22⋊C4—22no—
364C22⋊C4—22no—
365C22⋊C4—22no—
366C22⋊C4—5422no—
367C22⋊C4—5422no—
368C22⋊C4—5422no—
369C22⋊C4—5422no—
370C22⋊C4—5422no—
371C22⋊C4—5422no—
372C22⋊C4—5422no—
373C22⋊C4—10841no—
374C22⋊C4—10841no—
375C42—5422no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—2712no—
384C23—2712no—
385C23—2712no—
386C23—2712no—
387C23—5421no—
388C23—5421no—
389C23—5421no—
390C23—5421no—
391C23—5421no—
392C23—5421no—
393C23—5421no—
394C23—5421no—
395C23—5421no—
396C23—5421no—
397C23—5421no—
398C23—5421no—
399C23—5421no—
400C23—5421no—
401C23—5421no—
402C23—5421no—
403C23—5421no—
404C23C23×D27111yescharacteristic
405D4—5428no—
406D4—10844no—
407D4—10844no—
408D4—10844no—
409D4—10844no—
410D4—10844no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—2712no—
427C2×C4—2712no—
428C2×C4—2712no—
429C2×C4—2712no—
430C2×C4—5428no—
431C2×C4—5421no—
432C2×C4—5421no—
433C2×C4—5421no—
434C2×C4—5421no—
435C2×C4—5421no—
436C2×C4—5421no—
437C2×C4—5421no—
438C2×C4—5421no—
439C2×C4—5421no—
440C2×C4—5421no—
441C2×C4—5421no—
442C2×C4—10844no—
443C2×C4—10844no—
444C2×C4—10844no—
445C2×C4D4×D27112yes—
446C2×C4D4×D27112yes—
447C2×C4D4×D27112yes—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—2714no—
456C22—5422no—
457C22—5422no—
458C22—5422no—
459C22—5422no—
460C22—5422no—
461C22—5422no—
462C22—5422no—
463C22—5422no—
464C22—5422no—
465C22—10841no—
466C22—10841no—
467C22—10841no—
468C22—10841no—
469C22—10841no—
470C22—10841no—
471C22—10841no—
472C22—10841no—
473C22—10841no—
474C22—10841no—
475C22—10841no—
476C22C2×D4×D27111yescharacteristic
477C22C2×D4×D27111yescharacteristic
478C22C2×D4×D27111yescharacteristic
479C22#41×D27111yescharacteristic
480C22#41×D27111yescharacteristic
481C22#2257620611yescharacteristic
482C22#2257620811yescharacteristic
483C4—22no—
484C4—22no—
485C4—22no—
486C4—41no—
487C4—41no—
488C4—41no—
489C4—2714no—
490C4—5422no—
491C4—10841no—
492C4—10841no—
493C4—10841no—
494C2—81no—
495C2—5424no—
496C2—10842no—
497C2—21681no—
498C2#121×D27111yescharacteristic
499C2#121×D27111yescharacteristic
500C2#3177447711yescharacteristic
501C2#3177453111yescharacteristic
502C2#3177454911yescharacteristic
503C2#3177455711yescharacteristic
504C2#3177459811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
Fitting#396×C271C2
hypercenter#396D271
hypocenterC271#2416
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. C2×#22576135C2semidirect
2. C2×#22576133C2semidirect
3. C2×#22576133C2semidirect
4. C2×#22576133C2semidirect
5. C2×#22576122C2semidirect
6. C2×C4⋊C4×D271C2semidirect
7. #31773819C2semidirect
8. #31773814C2semidirect
9. #31773813C2semidirect
10. #31773797C2semidirect
11. #31773792C2semidirect
12. #31773775C2semidirect
13. #31773773C2semidirect
14. #31773756C2semidirect
15. #396×C271C2semidirect
68. C2×D1084D4semidirect
94. C2×C4×D271D4semidirect
99. Dic271⋊C4D4semidirect
229. C2×D271#121semidirect
240. Dic271#121semidirect
252. C271#2416semidirect
261. #396D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761352
C2×#225761332
C2×#225761222
C2×C4⋊C4×D2712
#317738192
#317738142
#317738132
#317737972
#317737922
#317737752
#317737732
#317737562
#396×C2712
#2416271