Finite Group Databasebeta

← #10698344

Subgroups — #10698344

Order 768.

466 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 95 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#73357C211yesmaximal, characteristic
2C2×C3×#328C211yesmaximal, characteristic
3#71701C211yesmaximal, characteristic
4#71690C211yesmaximal, characteristic
5#67678C211yesmaximal, characteristic
6#67676C211yesmaximal, characteristic
7#67612C211yesmaximal, characteristic
8#10375—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C2×#924—21no—
11C2×C3×C22⋊C8C2211yescharacteristic
12C2×C3×#95C2211yescharacteristic
13#5212C2211yescharacteristic
14#5212C2211yescharacteristic
15#5210—21no—
16C4×#217C2211yescharacteristic
17C2×#830C2211yescharacteristic
18#4657—21no—
19#4656—21no—
20#4647—21no—
21#4646—21no—
22C3×#328—21no—
23C3×#328—21no—
24#4538—21no—
25#4538—21no—
26#4526—21no—
27#4526—21no—
28#2549—31no—
29#1996—31no—
30#1520—31no—
31C2×#328D311yescharacteristic
32#1294—31no—
33#1289—31no—
34#1282—31no—
35C22×C3×Q8Q811yescharacteristic, complemented
36C23×C3×C4—21no—
37C23×C3×C4C2311yescharacteristic
38C22×Dic6—21no—
39C22×C3×C8—21no—
40C3×C22⋊Q8—21no—
41C3×C22⋊Q8—21no—
42C3×C22⋊Q8—21no—
43C3×C22⋊Q8—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4Q811yescharacteristic, complemented
46C2×C3×C22⋊C4—21no—
47C2×#217—21no—
48C2×#217—21no—
49C2×#217D411yescharacteristic
50C2×#217D411yescharacteristic
51C2×C4⋊Dic3—21no—
52C2×C4⋊Dic3—21no—
53C2×C4⋊Dic3—21no—
54#924—42no—
55#924—42no—
56C2×Dic3⋊C4—21no—
57C2×Dic3⋊C4—21no—
58C2×Dic3⋊C4—21no—
59C2×Dic3⋊C4—21no—
60C2×C4×Dic3—21no—
61C22×C3⋊C8—21no—
62C3×C22⋊C8—21no—
63C3×C22⋊C8—21no—
64C3×#95—21no—
65C3×#95D411yescharacteristic
66C3×#95D411yescharacteristic
67C3⋊#95—21no—
68C3⋊#95—21no—
69C3⋊#95—21no—
70C3⋊#95—21no—
71C3⋊#95D411yescharacteristic
72C3⋊#95D411yescharacteristic
73#830—21no—
74#830—21no—
75C2×C22⋊Q8—61no—
76C2×C22⋊Q8C2×D311yescharacteristic
77C2×C22⋊C8—31no—
78C2×C22⋊C8C2×D311yescharacteristic
79#398—61no—
80#395—61no—
81#391—61no—
82#389—61no—
83#385—61no—
84#380—31no—
85#380—31no—
86C4×C22⋊C4—31no—
87C2×#95C2×D311yescharacteristic
88#342—61no—
89#342—61no—
90#328—21no—
91#328—21no—
92#328—61no—
93#328—61no—
94C24×C3C2×D411yescharacteristic
95C2×C3×Q8—21no—
96C2×C3×Q8—21no—
97C2×C3×Q8—81no—
98C22×C3×C4—21no—
99C22×C3×C4—21no—
100C22×C3×C4—21no—
101C22×C3×C4—21no—
102C22×C3×C4—42no—
103C22×C3×C4—41no—
104C22×C3×C4—41no—
105C22×C3×C4—81no—
106C22×C3×C4—81no—
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×Q811yescharacteristic
110C22×C3×C4#4111yescharacteristic
111C22×C3×C4#4111yescharacteristic
112C22×C3×C4#4111yescharacteristic
113C22×Dic3—21no—
114C22×Dic3—21no—
115C22×Dic3—21no—
116C22×Dic3—21no—
117C22×Dic3—41no—
118C2×Dic6—42no—
119C2×Dic6—42no—
120C2×Dic6—81no—
121C2×C3×C8—41no—
122C2×C3×C8—81no—
123C3×C4⋊C4—21no—
124C3×C4⋊C4—21no—
125C3×C4⋊C4—41no—
126C3×C4⋊C4—41no—
127C3×C22⋊C4—41no—
128C3×C22⋊C4—41no—
129#217—21no—
130#217—21no—
131#217—41no—
132#217—41no—
133#217—41no—
134#217—41no—
135#217SD1612yes—
136#217Q1612yes—
137C4⋊Dic3—42no—
138C4⋊Dic3—81no—
139C4⋊Dic3—81no—
140Dic3⋊C4—42no—
141Dic3⋊C4—42no—
142Dic3⋊C4—42no—
143Dic3⋊C4—81no—
144C4×Dic3—81no—
145C2×C3⋊C8—41no—
146C2×C3⋊C8—81no—
147C22×Q8—61no—
148C22×Q8Dic611yescharacteristic, complemented
149C23×C4—21no—
150C23×C4C22×D311yescharacteristic
151C22×C8—21no—
152C22×C8—61no—
153C22⋊Q8—21no—
154C22⋊Q8—21no—
155C22⋊Q8—21no—
156C22⋊Q8—21no—
157C22⋊Q8—122no—
158C22⋊Q8—122no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—61no—
161C2×C4⋊C4—61no—
162C2×C4⋊C4—61no—
163C2×C4⋊C4—61no—
164C2×C4⋊C4—61no—
165C2×C4⋊C4—61no—
166C2×C4⋊C4—61no—
167C2×C4⋊C4Dic611yescharacteristic, complemented
168C2×C22⋊C4—21no—
169C2×C22⋊C4—31no—
170C2×C22⋊C4—31no—
171C2×C22⋊C4—61no—
172C2×C22⋊C4—61no—
173C2×C42—61no—
174C22⋊C8—21no—
175C22⋊C8—21no—
176C22⋊C8—61no—
177C22⋊C8—61no—
178#95—21no—
179#95—31no—
180#95—31no—
181#95—61no—
182#95—61no—
183#95—61no—
184#95—61no—
185#95#6711yescharacteristic
186#95#6711yescharacteristic
187C23×C3—41no—
188C23×C3—41no—
189C23×C3#12011yescharacteristic
190C23×C3C22⋊Q811yescharacteristic
191C23×C3#12411yescharacteristic
192C23×C3C2×SD1611yescharacteristic
193C23×C3C2×Q1611yescharacteristic
194C23×C3AGL(1,ℤ/8)11yescharacteristic
195C23×C3#13711yescharacteristic
196C3×Q8—81no—
197C3×Q8—81no—
198C2×C3×C4—22no—
199C2×C3×C4—41no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4—82no—
213C2×C3×C4—82no—
214C2×C3×C4—81no—
215C2×C3×C4—81no—
216C2×C3×C4C22⋊Q811yescharacteristic
217C2×C3×C4C22⋊Q811yescharacteristic
218C2×C3×C4#12411yescharacteristic
219C2×C3×C4#12411yescharacteristic
220C2×Dic3—22no—
221C2×Dic3—22no—
222C2×Dic3—41no—
223C2×Dic3—41no—
224C2×Dic3—41no—
225C2×Dic3—41no—
226C2×Dic3—41no—
227C2×Dic3—41no—
228C2×Dic3—41no—
229C2×Dic3—41no—
230C2×Dic3—41no—
231C2×Dic3—41no—
232C2×Dic3—81no—
233C2×Dic3—81no—
234C2×Dic3—81no—
235Dic6—84no—
236C3×C8—81no—
237C3⋊C8—81no—
238C24D3×D411yescharacteristic
239C2×Q8—21no—
240C2×Q8—21no—
241C2×Q8—81no—
242C2×Q8—122no—
243C2×Q8—122no—
244C2×Q8—241no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—42no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—61no—
253C22×C4—61no—
254C22×C4—61no—
255C22×C4—61no—
256C22×C4—81no—
257C22×C4—81no—
258C22×C4—121no—
259C22×C4C2×Dic611yescharacteristic
260C22×C4#23511yescharacteristic
261C22×C4D3×D411yescharacteristic
262C22×C4#23711yescharacteristic
263C22×C4#23711yescharacteristic
264C22×C4C2×#6711yescharacteristic
265C2×C8—41no—
266C2×C8—81no—
267C2×C8—121no—
268C2×C8—241no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—122no—
274C4⋊C4—122no—
275C4⋊C4—122no—
276C4⋊C4—122no—
277C4⋊C4—241no—
278C4⋊C4—241no—
279C4⋊C4—241no—
280C22⋊C4—32no—
281C22⋊C4—32no—
282C22⋊C4—41no—
283C22⋊C4—41no—
284C22⋊C4—61no—
285C22⋊C4—61no—
286C22⋊C4—121no—
287C22⋊C4—121no—
288C22⋊C4—121no—
289C22⋊C4—121no—
290C42—241no—
291C22×C3—21no—
292C22×C3—21no—
293C22×C3—21no—
294C22×C3—21no—
295C22×C3—21no—
296C22×C3—21no—
297C22×C3—41no—
298C22×C3—41no—
299C22×C3—41no—
300C22×C3—41no—
301C22×C3#39311yescharacteristic
302C22×C3#45011yescharacteristic
303C22×C3#45111yescharacteristic
304C22×C3#47511yescharacteristic
305C22×C3#47711yescharacteristic
306C22×C3#48711yescharacteristic
307C22×C3#48911yescharacteristic
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—42no—
313C3×C4—81no—
314C3×C4—81no—
315C3×C4—81no—
316Dic3—42no—
317Dic3—42no—
318Dic3—81no—
319Dic3—81no—
320Dic3—161no—
321C23—41no—
322C23—41no—
323C23#87811yescharacteristic
324C23#88511yescharacteristic
325C23D3×SD1611yescharacteristic
326C23#91411yescharacteristic
327C23D3×Q1611yescharacteristic
328C23#91811yescharacteristic
329C23#93711yescharacteristic
330Q8—81no—
331Q8—81no—
332Q8—244no—
333C2×C4—22no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—62no—
348C2×C4—62no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—121no—
354C2×C4—121no—
355C2×C4—121no—
356C2×C4—121no—
357C2×C4—121no—
358C2×C4—121no—
359C2×C4—121no—
360C2×C4—121no—
361C2×C4—121no—
362C2×C4—121no—
363C2×C4—241no—
364C2×C4—241no—
365C2×C4—241no—
366C2×C4#87811yescharacteristic
367C2×C4#88511yescharacteristic
368C2×C4#92411yescharacteristic
369C2×C4#93611yescharacteristic
370C8—81no—
371C8—241no—
372C2×C3—21no—
373C2×C3—21no—
374C2×C3—21no—
375C2×C3—21no—
376C2×C3—21no—
377C2×C3—21no—
378C2×C3#202411yescharacteristic
379C2×C3#203111yescharacteristic
380C2×C3#203211yescharacteristic
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22#479111yescharacteristic
392C22#479211yescharacteristic
393C22#479311yescharacteristic
394C22#479411yescharacteristic
395C22#494711yescharacteristic
396C22#516911yescharacteristic
397C22#518611yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—42no—
403C4—81no—
404C4—81no—
405C4—81no—
406C4—122no—
407C4—122no—
408C4—241no—
409C4—241no—
410C4—481no—
411C3#1037511yescharacteristic, complemented, Sylow
412C2—21no—
413C2—21no—
414C2—21no—
415C2—21no—
416C2—21no—
417C2—21no—
418C2#7002511yescharacteristic
419C2#7023111yescharacteristic
420C2#7036811yescharacteristic
subgroupG/H
centerC22#4947
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC2×C3×#328C2
socleC22×C3#393
prime radicalC24×C3C2×D4
hypercenterC2×#328D3
hypocenterC3#10375
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#328

Complements

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

NH ≅ G/Ntype
35. C22×C3×Q8Q8semidirect
45. C2×C3×C4⋊C4Q8semidirect
148. C22×Q8Dic6semidirect
167. C2×C4⋊C4Dic6semidirect
411. C3#10375semidirect

Maximal subgroups

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

subgroupindex
#733572
C2×C3×#3282
#717012
#716902
#676782
#676762
#676122
#103753