Finite Group Databasebeta

← #10898320

Subgroups — #10898320

Order 768.

708 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#73430C211yesmaximal, characteristic, complemented
2#73415C211yesmaximal, characteristic, complemented
3C3×#1768C211yesmaximal, characteristic, complemented
4#70484C211yesmaximal, characteristic, complemented
5#70484C211yesmaximal, characteristic, complemented
6#70481C211yesmaximal, characteristic
7#70463C211yesmaximal, characteristic, complemented
8#69818C211yesmaximal, characteristic, complemented
9#69818C211yesmaximal, characteristic, complemented
10#69776C211yesmaximal, characteristic
11#69770C211yesmaximal, characteristic, complemented
12#68673C211yesmaximal, characteristic
13#68670C211yesmaximal, characteristic, complemented
14#68625C211yesmaximal, characteristic, complemented
15#68624C211yesmaximal, characteristic, complemented
16#29436—31nomaximal, Sylow
17C22×D24C2211yescharacteristic
18C2×C4×D12C2211yescharacteristic
19C2×C3×C4⋊C8C2211yescharacteristic
20C2×C3×C4⋊C8C2211yescharacteristic
21C2×C3×#97C2211yescharacteristic
22C2×C3×C4×C8C2211yescharacteristic
23C3×#376C2211yescharacteristic
24C2×#821C2211yescharacteristic
25C2×#821C2211yescharacteristic
26C2×#821C2211yescharacteristic
27C2×#821C2211yescharacteristic
28C2×#821C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×#818C2211yescharacteristic
31C2×#818C2211yescharacteristic
32C2×#818C2211yescharacteristic
33#4939C2211yescharacteristic
34#4939C2211yescharacteristic
35#4939C2211yescharacteristic
36#4939C2211yescharacteristic
37#4939C2211yescharacteristic
38C4×C4⋊Dic3C2211yescharacteristic
39#4928C2211yescharacteristic
40#4928C2211yescharacteristic
41#4928C2211yescharacteristic
42#4928C2211yescharacteristic
43#4928C2211yescharacteristic
44#4676—21no—
45#4675—21no—
46#4675—21no—
47#4675—21no—
48#4669—21no—
49#4669—21no—
50#4667—21no—
51#4664—21no—
52C3×#336C2211yescharacteristic
53C3×#336C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#2644—31no—
61#2319—31no—
62#2087—31no—
63#2067—31no—
64#2067—31no—
65#2054—31no—
66#1970—31no—
67#1942—31no—
68#1936—31no—
69#1936—31no—
70#1925—31no—
71#1921—31no—
72#1879—31no—
73#1862—31no—
74#1768D311yescharacteristic, complemented
75C22×D12C2311yescharacteristic
76C22×D12C2311yescharacteristic
77C22×C4×D3—21no—
78C22×C3×C8C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C2×C3×C42C2311yescharacteristic
83C2×C3×C42C2311yescharacteristic
84C2×C3×C42C2311yescharacteristic
85C2×(C2×D3)⋊C4—21no—
86C2×(C2×D3)⋊C4—21no—
87C2×(C2×D3)⋊C4—21no—
88C2×(C2×D3)⋊C4—21no—
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×C4⋊Dic3C2311yescharacteristic
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×C4×Dic3—21no—
108C2×D24—22no—
109C2×D24—22no—
110C2×D24—41no—
111C4×D12—22no—
112C4×D12—22no—
113C3×C4⋊C8—22no—
114C3×C4⋊C8—22no—
115C3×#97—41no—
116C3×C4×C8—41no—
117C3×#95D411yescharacteristic
118C3×#95D411yescharacteristic
119C3×#95D411yescharacteristic
120C3×#95D411yescharacteristic
121C3⋊#95—21no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127#821—22no—
128#821—22no—
129#821—22no—
130#821—22no—
131#821—41no—
132#821—41no—
133#818—22no—
134#818—22no—
135#818—41no—
136C22×D8—31no—
137C2×C4×D4—31no—
138C2×#107—31no—
139C2×#107—31no—
140C2×#107—31no—
141C2×C4⋊C8C2×D311yescharacteristic
142C2×C4⋊C8C2×D311yescharacteristic
143C2×#102—31no—
144C2×#102—31no—
145C2×#102—31no—
146C2×#102—31no—
147C2×#102—31no—
148C2×#102—31no—
149C2×#97C2×D311yescharacteristic
150C2×C4×C8C2×D311yescharacteristic
151#397—61no—
152#397—61no—
153#397—61no—
154#396—61no—
155#390—31no—
156#390—31no—
157#390—31no—
158#390—31no—
159#390—31no—
160#389—31no—
161#389—31no—
162#389—31no—
163#389—31no—
164#389—31no—
165#388—61no—
166#388—61no—
167#381—61no—
168#380—61no—
169C4×C4⋊C4—31no—
170#376C2×D311yescharacteristic
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340—31no—
177#336C2×D311yescharacteristic
178#336C2×D311yescharacteristic
179C23×D3—21no—
180C23×D3—21no—
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2411yescharacteristic
188C22×Dic3—21no—
189C22×Dic3—21no—
190C22×Dic3—21no—
191C22×Dic3—21no—
192C22×Dic3—21no—
193C22×Dic3—21no—
194C2×D12—21no—
195C2×D12—21no—
196C2×D12—21no—
197C2×D12—21no—
198C2×D12—21no—
199C2×D12—21no—
200C2×D12—21no—
201C2×D12—21no—
202C2×D12—21no—
203C2×D12—21no—
204C2×D12—21no—
205C2×D12#4112yes—
206C2×C4×D3—24no—
207C2×C4×D3—81no—
208C2×C4×D3—81no—
209C2×C3×C8—21no—
210C2×C3×C8—21no—
211C2×C3×C8—21no—
212C2×C3×C8—21no—
213C2×C3×C8—21no—
214C2×C3×C8—21no—
215C2×C3×C8—21no—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8#4112yes—
220C2×C3×C8#4112yes—
221C3×C42—21no—
222C3×C42—21no—
223C3×C42—21no—
224C3×C42—21no—
225C3×C42—21no—
226C3×C42—21no—
227(C2×D3)⋊C4—24no—
228(C2×D3)⋊C4—81no—
229(C2×D3)⋊C4—81no—
230(C2×D3)⋊C4—81no—
231(C2×D3)⋊C4—81no—
232(C2×D3)⋊C4—81no—
233(C2×D3)⋊C4—81no—
234(C2×D3)⋊C4—81no—
235(C2×D3)⋊C4—81no—
236(C2×D3)⋊C4—81no—
237(C2×D3)⋊C4—81no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3#4112yes—
250Dic3⋊C4—81no—
251Dic3⋊C4—81no—
252Dic3⋊C4—81no—
253Dic3⋊C4—81no—
254Dic3⋊C4—81no—
255C4×Dic3—81no—
256D24—44no—
257C22×D4—31no—
258C22×D4—31no—
259C23×C4—61no—
260C2×D8—62no—
261C2×D8—62no—
262C2×D8—121no—
263C22×C8C22×D311yescharacteristic
264C22×C8C22×D311yescharacteristic
265C22×C8C22×D311yescharacteristic
266C22×C8C22×D311yescharacteristic
267C4×D4—62no—
268C4×D4—62no—
269C2×C4⋊C4—31no—
270C2×C4⋊C4—31no—
271C2×C4⋊C4—31no—
272C2×C4⋊C4—31no—
273C2×C4⋊C4—31no—
274C2×C4⋊C4—31no—
275C2×C4⋊C4—61no—
276C2×C4⋊C4—61no—
277C2×C4⋊C4—61no—
278C2×C4⋊C4—61no—
279C2×C4⋊C4—61no—
280C2×C22⋊C4—61no—
281C2×C22⋊C4—61no—
282C2×C22⋊C4—61no—
283C2×C22⋊C4—61no—
284C2×C22⋊C4—61no—
285C2×C22⋊C4—61no—
286C2×C22⋊C4—61no—
287C2×C22⋊C4—61no—
288C2×C22⋊C4—61no—
289C2×C22⋊C4—61no—
290C2×C22⋊C4—61no—
291C2×C42—61no—
292C2×C42C22×D311yescharacteristic
293C2×C42C22×D311yescharacteristic
294C2×C42C22×D311yescharacteristic
295#107—62no—
296#107—62no—
297#107—121no—
298C4⋊C8—22no—
299C4⋊C8—22no—
300#102—62no—
301#102—62no—
302#102—62no—
303#102—62no—
304#102—121no—
305#102—121no—
306#97—41no—
307C4×C8—41no—
308#95—61no—
309#95—61no—
310#95—61no—
311#95—61no—
312#95—61no—
313#95—61no—
314#95D1211yescharacteristic
315#95D1211yescharacteristic
316#95D1211yescharacteristic
317#95D1211yescharacteristic
318C23×C3C22×D411yescharacteristic
319C22×D3—22no—
320C22×D3—41no—
321C22×D3—41no—
322C22×D3—41no—
323C22×D3—41no—
324C22×D3—41no—
325C22×D3—41no—
326C22×D3—41no—
327C22×D3—41no—
328C22×D3—41no—
329C22×D3—41no—
330C22×D3—41no—
331C22×D3—41no—
332C22×D3—41no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4C2×#4111yescharacteristic
352C2×C3×C4C2×#4111yescharacteristic
353C2×C3×C4C2×#4111yescharacteristic
354C2×C3×C4C2×#4111yescharacteristic
355C2×C3×C4H(5,𝔽2)11yescharacteristic
356C2×C3×C42−1+411yescharacteristic
357C2×Dic3—22no—
358C2×Dic3—41no—
359C2×Dic3—41no—
360C2×Dic3—41no—
361C2×Dic3—41no—
362C2×Dic3—41no—
363C2×Dic3—41no—
364C2×Dic3—41no—
365C2×Dic3—41no—
366C2×Dic3—41no—
367C2×Dic3—41no—
368C2×Dic3—41no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375D12—22no—
376D12—22no—
377D12—41no—
378D12—41no—
379D12—41no—
380D12—41no—
381D12—41no—
382D12—41no—
383C4×D3—84no—
384C3×C8—22no—
385C3×C8—22no—
386C3×C8—41no—
387C3×C8—41no—
388C24—61no—
389C24—61no—
390C2×D4—32no—
391C2×D4—61no—
392C2×D4—61no—
393C2×D4—61no—
394C2×D4—61no—
395C2×D4—61no—
396C2×D4—61no—
397C2×D4—61no—
398C2×D4—61no—
399C2×D4—61no—
400C2×D4—61no—
401C2×D4—61no—
402C22×C4—64no—
403C22×C4—61no—
404C22×C4—61no—
405C22×C4—61no—
406C22×C4—61no—
407C22×C4—61no—
408C22×C4—61no—
409C22×C4—241no—
410C22×C4—241no—
411C22×C4C2×D1211yescharacteristic
412C22×C4C2×D1211yescharacteristic
413C22×C4C2×D1211yescharacteristic
414C22×C4C2×D1211yescharacteristic
415C22×C4C2×D1211yescharacteristic
416C22×C4C2×D1211yescharacteristic
417C22×C4C23×D311yescharacteristic
418D8—124no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8#23512yes—
430C2×C8#23512yes—
431C4⋊C4—32no—
432C4⋊C4—61no—
433C4⋊C4—61no—
434C4⋊C4—61no—
435C4⋊C4—61no—
436C4⋊C4—61no—
437C4⋊C4—61no—
438C4⋊C4—61no—
439C4⋊C4—61no—
440C4⋊C4—61no—
441C4⋊C4—61no—
442C4⋊C4—61no—
443C4⋊C4—241no—
444C4⋊C4—241no—
445C4⋊C4—241no—
446C4⋊C4—241no—
447C4⋊C4—241no—
448C22⋊C4—64no—
449C22⋊C4—241no—
450C22⋊C4—241no—
451C22⋊C4—241no—
452C22⋊C4—241no—
453C22⋊C4—241no—
454C22⋊C4—241no—
455C22⋊C4—241no—
456C22⋊C4—241no—
457C22⋊C4—241no—
458C22⋊C4—241no—
459C42—21no—
460C42—21no—
461C42—21no—
462C42—21no—
463C42—21no—
464C42—21no—
465C42—241no—
466C22×C3#52511yescharacteristic
467C22×C3#53211yescharacteristic
468C22×C3#53711yescharacteristic
469C22×C3#57611yescharacteristic
470C22×C3#57611yescharacteristic
471C22×C3#57611yescharacteristic
472C22×C3#57811yescharacteristic
473C2×D3—42no—
474C2×D3—42no—
475C2×D3—42no—
476C2×D3—81no—
477C2×D3—81no—
478C2×D3—81no—
479C2×D3—81no—
480C2×D3—81no—
481C2×D3—81no—
482C2×D3—81no—
483C2×D3—81no—
484C2×D3—81no—
485C2×D3—81no—
486C2×D3—81no—
487C3×C4—41no—
488C3×C4—41no—
489C3×C4—41no—
490C3×C4—41no—
491C3×C4—41no—
492C3×C4—41no—
493C3×C4#55111yescharacteristic
494C3×C4#55211yescharacteristic
495C3×C4#55311yescharacteristic
496C3×C4#55611yescharacteristic
497Dic3—42no—
498Dic3—81no—
499Dic3—81no—
500Dic3—81no—
501Dic3—81no—
502Dic3—81no—
503C23—62no—
504C23—121no—
505C23—121no—
506C23—121no—
507C23—121no—
508C23—121no—
509C23—121no—
510C23—121no—
511C23—121no—
512C23—121no—
513C23—121no—
514C23—121no—
515C23—121no—
516C23—121no—
517C23C22×D1211yescharacteristic
518D4—62no—
519D4—62no—
520D4—121no—
521D4—121no—
522D4—121no—
523D4—121no—
524D4—121no—
525D4—121no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—62no—
545C2×C4—121no—
546C2×C4—121no—
547C2×C4—121no—
548C2×C4—121no—
549C2×C4—121no—
550C2×C4—121no—
551C2×C4—121no—
552C2×C4—121no—
553C2×C4—121no—
554C2×C4—121no—
555C2×C4—121no—
556C2×C4—121no—
557C2×C4—121no—
558C2×C4—121no—
559C2×C4—121no—
560C2×C4—121no—
561C2×C4—121no—
562C2×C4—244no—
563C2×C4C2×#23511yescharacteristic
564C2×C4C2×#23511yescharacteristic
565C2×C4#100411yescharacteristic
566C2×C4#100711yescharacteristic
567C2×C4D3×#4111yescharacteristic
568C2×C4D3×#4111yescharacteristic
569C8—22no—
570C8—22no—
571C8—41no—
572C8—41no—
573C2×C3#260111yescharacteristic
574C2×C3#312211yescharacteristic
575C2×C3#312411yescharacteristic
576C2×C3#317511yescharacteristic
577C2×C3#317611yescharacteristic
578C2×C3#325911yescharacteristic
579C2×C3#326211yescharacteristic
580D3—82no—
581D3—161no—
582C22—122no—
583C22—122no—
584C22—122no—
585C22—241no—
586C22—241no—
587C22—241no—
588C22—241no—
589C22—241no—
590C22—241no—
591C22—241no—
592C22—241no—
593C22—241no—
594C22—241no—
595C22—241no—
596C22#547711yescharacteristic
597C22#551011yescharacteristic
598C22#552411yescharacteristic
599C22#575211yescharacteristic
600C22#575211yescharacteristic
601C22#575211yescharacteristic
602C22#575311yescharacteristic
603C4—41no—
604C4—41no—
605C4—41no—
606C4—41no—
607C4—41no—
608C4—41no—
609C4—122no—
610C4—241no—
611C4—241no—
612C4—241no—
613C4—241no—
614C4—241no—
615C4#554011yescharacteristic
616C4#555611yescharacteristic
617C4#556811yescharacteristic
618C4#558111yescharacteristic
619C3#2943611yescharacteristic, complemented, Sylow
620C2—242no—
621C2—481no—
622C2#7381311yescharacteristic
623C2#7856811yescharacteristic
624C2#7856911yescharacteristic
625C2#7909511yescharacteristic
626C2#7909711yescharacteristic
627C2#7938611yescharacteristic
628C2#7938811yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1768C2
socleC23×C3C22×D4
prime radicalC22×D24C22
hypercenter#1768D3
hypocenterC3#29436
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ #376 ◁ #1768

Complements

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

NH ≅ G/Ntype
1. #73430C2semidirect
2. #73415C2semidirect
3. C3×#1768C2semidirect
4. #70484C2semidirect
5. #70484C2semidirect
7. #70463C2semidirect
8. #69818C2semidirect
9. #69818C2semidirect
11. #69770C2semidirect
13. #68670C2semidirect
14. #68625C2semidirect
15. #68624C2semidirect
74. #1768D3semidirect
619. C3#29436semidirect

Maximal subgroups

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

subgroupindex
#734302
#734152
C3×#17682
#704842
#704812
#704632
#698182
#697762
#697702
#686732
#686702
#686252
#686242
#294363