Finite Group Databasebeta

← #10698417

Subgroups — #10698417

Order 768.

602 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#73553C211yesmaximal, characteristic, complemented
2C3×#1789C211yesmaximal, characteristic
3#72139C211yesmaximal, characteristic
4#72138C211yesmaximal, characteristic
5#67686C211yesmaximal, characteristic, complemented
6#67682C211yesmaximal, characteristic, complemented
7#67616C211yesmaximal, characteristic, complemented
8#10386—31nomaximal, Sylow
9C22×C3×#41C2211yescharacteristic
10C2×C3×#104—21no—
11C2×C3×#104—21no—
12C3×#407C2211yescharacteristic
13C3×C4×C22⋊C4C2211yescharacteristic
14C2×#837—21no—
15C2×#837—21no—
16C2×#834—21no—
17C2×#834—21no—
18#5211C2211yescharacteristic
19#5211C2211yescharacteristic
20C4×#217C2211yescharacteristic
21#5207C2211yescharacteristic
22#4949—21no—
23#4947—21no—
24#4933—21no—
25#4929—21no—
26#4654—21no—
27#4646—21no—
28#2607—31no—
29#1789—31no—
30#1789D311yescharacteristic
31#1782—31no—
32#1306—31no—
33#1284—31no—
34#1284—31no—
35C2×C3×#41—21no—
36C2×C3×#41—21no—
37C2×C3×#41—21no—
38C2×C3×#41—21no—
39C2×C3×#41—42no—
40C2×C3×#41—42no—
41C22×C3×Q8Q811yescharacteristic, complemented
42C22×C3×D4—21no—
43C22×C3×D4Q811yescharacteristic, complemented
44C23×C3×C4—21no—
45C23×C3×C4C2311yescharacteristic
46C22×Dic6—21no—
47C2×C3×M16—21no—
48C2×C3×C22⋊C4—21no—
49C2×C3×C42D411yescharacteristic
50C2×C3×C42D411yescharacteristic
51C2×#217—21no—
52C2×#217—21no—
53C2×#217D411yescharacteristic
54C2×#217D411yescharacteristic
55C2×C4⋊Dic3—21no—
56C2×Dic3⋊C4—21no—
57C2×Dic3⋊C4—21no—
58C2×C4×Dic3D411yescharacteristic
59C2×C4×Dic3D411yescharacteristic
60C2×#208—21no—
61C3×#104—42no—
62C3×#104—42no—
63C3×C22⋊C8—21no—
64C3×C22⋊C8—21no—
65C3×#95—21no—
66#837—42no—
67#837—42no—
68#834—42no—
69#834—42no—
70C3⋊#95—21no—
71C3⋊#95—21no—
72C3⋊#95—21no—
73C3⋊#95—21no—
74C3⋊#95—21no—
75C3⋊#95—21no—
76C3⋊#95—21no—
77C3⋊#95—21no—
78C3⋊#95—21no—
79#830—21no—
80#830—21no—
81C22×#41C2×D311yescharacteristic
82C2×#104—21no—
83C2×#104—21no—
84C2×#104—61no—
85C2×#104—61no—
86C2×#99—61no—
87C2×#99—61no—
88#407—31no—
89#407C2×D311yescharacteristic
90#399—61no—
91#393—61no—
92#391—61no—
93#391—61no—
94#387—61no—
95#387—61no—
96#381—31no—
97#381—31no—
98C4×C22⋊C4—31no—
99C4×C22⋊C4C2×D311yescharacteristic
100C24×C3—21no—
101C24×C3C2×D411yescharacteristic
102C3×#41—22no—
103C3×#41—22no—
104C3×#41—22no—
105C3×#41—22no—
106C3×#41—41no—
107C3×#41—41no—
108C3×#41—41no—
109C3×#41—41no—
110C3×#41—42no—
111C3×#41—42no—
112C3×#41—42no—
113C3×#41—42no—
114C2×C3×Q8—21no—
115C2×C3×Q8—21no—
116C2×C3×Q8—42no—
117C2×C3×D4—22no—
118C2×C3×D4—22no—
119C2×C3×D4—21no—
120C2×C3×D4—21no—
121C2×C3×D4—42no—
122C2×C3×D4—41no—
123C2×C3×D4—41no—
124C2×C3×D4—41no—
125C2×C3×D4—41no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—21no—
129C22×C3×C4—41no—
130C22×C3×C4—41no—
131C22×C3×C4—42no—
132C22×C3×C4—42no—
133C22×C3×C4—41no—
134C22×C3×C4—41no—
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×Q811yescharacteristic
138C22×C3×C4#4111yescharacteristic
139C22×C3×C4#4111yescharacteristic
140C22×C3×C4#4111yescharacteristic
141C22×Dic3—21no—
142C22×Dic3—21no—
143C22×Dic3—21no—
144C22×Dic3—21no—
145C22×Dic3—41no—
146C2×Dic6—42no—
147C2×Dic6—42no—
148C2×Dic6—81no—
149C3×M16—41no—
150C3×M16—41no—
151C2×C3×C8—41no—
152C3×C22⋊C4—42no—
153C3×C42—41no—
154C3×C42—41no—
155#217—42no—
156#217—41no—
157#217—41no—
158#217—41no—
159#217—41no—
160C4⋊Dic3—42no—
161Dic3⋊C4—42no—
162Dic3⋊C4—81no—
163C4×Dic3—41no—
164C4×Dic3—41no—
165#208—41no—
166#208—41no—
167C2×C3⋊C8—41no—
168C2×#41—21no—
169C2×#41—21no—
170C2×#41—21no—
171C2×#41—21no—
172C2×#41—42no—
173C2×#41—42no—
174C22×Q8—61no—
175C22×Q8Dic611yescharacteristic, complemented
176C22×D4—21no—
177C22×D4Dic611yescharacteristic, complemented
178C23×C4—21no—
179C23×C4C22×D311yescharacteristic
180C2×M16—21no—
181C2×M16—61no—
182C2×C4⋊C4—61no—
183C2×C4⋊C4—61no—
184C2×C4⋊C4—61no—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—31no—
187C2×C22⋊C4—31no—
188C2×C22⋊C4—61no—
189C2×C22⋊C4—61no—
190C2×C42—31no—
191C2×C42—31no—
192C2×C42#6711yescharacteristic
193C2×C42#6711yescharacteristic
194#104—42no—
195#104—42no—
196#104—122no—
197#104—122no—
198#99—122no—
199#99—122no—
200C22⋊C8—21no—
201C22⋊C8—21no—
202C22⋊C8—61no—
203C22⋊C8—61no—
204#95—21no—
205#95—61no—
206#95—61no—
207#95—61no—
208#95—61no—
209#95—61no—
210#95—61no—
211#95—61no—
212#95—61no—
213#95—61no—
214C23×C3—21no—
215C23×C3—21no—
216C23×C3—22no—
217C23×C3—41no—
218C23×C3—41no—
219C23×C3—41no—
220C23×C3—41no—
221C23×C3—81no—
222C23×C3—81no—
223C23×C3#12011yescharacteristic
224C23×C3C22⋊Q811yescharacteristic
225C23×C3#12411yescharacteristic
226C3×Q8—22no—
227C3×Q8—22no—
228C3×Q8—41no—
229C3×Q8—41no—
230C3×D4—22no—
231C3×D4—22no—
232C3×D4—41no—
233C3×D4—41no—
234C3×D4—42no—
235C3×D4—42no—
236C3×D4—42no—
237C3×D4—42no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—22no—
243C2×C3×C4—22no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—42no—
249C2×C3×C4—42no—
250C2×C3×C4—42no—
251C2×C3×C4—42no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—41no—
255C2×C3×C4—41no—
256C2×C3×C4—41no—
257C2×C3×C4—41no—
258C2×C3×C4—41no—
259C2×C3×C4C22⋊Q811yescharacteristic
260C2×C3×C4C22⋊Q811yescharacteristic
261C2×C3×C4#12411yescharacteristic
262C2×C3×C4#12411yescharacteristic
263C2×Dic3—22no—
264C2×Dic3—41no—
265C2×Dic3—41no—
266C2×Dic3—41no—
267C2×Dic3—41no—
268C2×Dic3—41no—
269C2×Dic3—41no—
270C2×Dic3—41no—
271C2×Dic3—41no—
272C2×Dic3—41no—
273C2×Dic3—41no—
274C2×Dic3—41no—
275C2×Dic3—81no—
276C2×Dic3—81no—
277C2×Dic3—81no—
278Dic6—84no—
279C3×C8—81no—
280C3⋊C8—81no—
281C24—21no—
282C24D3×D411yescharacteristic
283#41—22no—
284#41—22no—
285#41—22no—
286#41—22no—
287#41—41no—
288#41—41no—
289#41—41no—
290#41—41no—
291#41—42no—
292#41—42no—
293#41—42no—
294#41—42no—
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8—42no—
298C2×Q8—122no—
299C2×Q8—122no—
300C2×Q8—241no—
301C2×D4—22no—
302C2×D4—22no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—42no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—61no—
320C22×C4—61no—
321C22×C4—61no—
322C22×C4—61no—
323C22×C4—121no—
324C22×C4C2×Dic611yescharacteristic
325C22×C4#23511yescharacteristic
326C22×C4D3×D411yescharacteristic
327C22×C4#23711yescharacteristic
328C22×C4#23711yescharacteristic
329C22×C4C2×#6711yescharacteristic
330M16—41no—
331M16—41no—
332M16—121no—
333M16—121no—
334C2×C8—41no—
335C2×C8—121no—
336C4⋊C4—122no—
337C4⋊C4—122no—
338C4⋊C4—241no—
339C22⋊C4—42no—
340C22⋊C4—122no—
341C22⋊C4—121no—
342C22⋊C4—121no—
343C22⋊C4—121no—
344C22⋊C4—121no—
345C42—41no—
346C42—41no—
347C42—121no—
348C42—121no—
349C22×C3—21no—
350C22×C3—21no—
351C22×C3—21no—
352C22×C3—21no—
353C22×C3—21no—
354C22×C3—21no—
355C22×C3—21no—
356C22×C3—21no—
357C22×C3—42no—
358C22×C3—81no—
359C22×C3—81no—
360C22×C3—82no—
361C22×C3—82no—
362C22×C3—81no—
363C22×C3—81no—
364C22×C3#39311yescharacteristic
365C22×C3B(4,𝔽2)11yescharacteristic
366C22×C3B(4,𝔽2)11yescharacteristic
367C3×C4—21no—
368C3×C4—21no—
369C3×C4—21no—
370C3×C4—21no—
371C3×C4—42no—
372C3×C4—42no—
373C3×C4—81no—
374Dic3—42no—
375Dic3—81no—
376Dic3—81no—
377Dic3—81no—
378Dic3—161no—
379C23—21no—
380C23—21no—
381C23—22no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—81no—
387C23—81no—
388C23#87811yescharacteristic
389C23#88511yescharacteristic
390C23#93711yescharacteristic
391Q8—22no—
392Q8—22no—
393Q8—41no—
394Q8—41no—
395Q8—244no—
396D4—22no—
397D4—22no—
398D4—41no—
399D4—41no—
400D4—42no—
401D4—42no—
402D4—42no—
403D4—42no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—62no—
426C2×C4—121no—
427C2×C4—121no—
428C2×C4—121no—
429C2×C4—121no—
430C2×C4—121no—
431C2×C4—121no—
432C2×C4—121no—
433C2×C4—121no—
434C2×C4—121no—
435C2×C4—121no—
436C2×C4—121no—
437C2×C4—241no—
438C2×C4—241no—
439C2×C4—241no—
440C2×C4#87811yescharacteristic
441C2×C4#88511yescharacteristic
442C2×C4#92411yescharacteristic
443C2×C4#93611yescharacteristic
444C8—81no—
445C8—241no—
446C2×C3—21no—
447C2×C3—21no—
448C2×C3—41no—
449C2×C3—41no—
450C2×C3—82no—
451C2×C3#202911yescharacteristic
452C2×C3#203111yescharacteristic
453C2×C3#203111yescharacteristic
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—42no—
463C22—81no—
464C22—81no—
465C22—82no—
466C22—82no—
467C22—81no—
468C22—81no—
469C22#494711yescharacteristic
470C22#503211yescharacteristic
471C22#503211yescharacteristic
472C4—21no—
473C4—21no—
474C4—21no—
475C4—21no—
476C4—42no—
477C4—42no—
478C4—81no—
479C4—122no—
480C4—241no—
481C4—241no—
482C4—241no—
483C4—481no—
484C3#1038611yescharacteristic, complemented, Sylow
485C2—21no—
486C2—21no—
487C2—41no—
488C2—41no—
489C2—82no—
490C2#6949711yescharacteristic
491C2#7036811yescharacteristic
492C2#7039411yescharacteristic
subgroupG/H
centerC22#4947
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1789C2
socleC22×C3#393
prime radicalC22×C3×#41C22
hypercenter#1789D3
hypocenterC3#10386
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1789

Complements

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

NH ≅ G/Ntype
1. #73553C2semidirect
5. #67686C2semidirect
6. #67682C2semidirect
7. #67616C2semidirect
41. C22×C3×Q8Q8semidirect
43. C22×C3×D4Q8semidirect
175. C22×Q8Dic6semidirect
177. C22×D4Dic6semidirect
484. C3#10386semidirect

Maximal subgroups

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

subgroupindex
#735532
C3×#17892
#721392
#721382
#676862
#676822
#676162
#103863