Finite Group Databasebeta

← #10695655

Subgroups — #10695655

Order 768.

594 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 99 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#82630C211yesmaximal, characteristic
2C2×#5033C211yesmaximal, characteristic
3#73518C211yesmaximal, characteristic, complemented
4C3×#1964C211yesmaximal, characteristic
5#69510C211yesmaximal, characteristic, complemented
6#69501C211yesmaximal, characteristic, complemented
7#67617C211yesmaximal, characteristic, complemented
8#9953—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C22×C4⋊Dic3C2211yescharacteristic
11C2×#924C2211yescharacteristic
12C2×C3×#99—22no—
13C3×#380C2211yescharacteristic
14C3×C4×C22⋊C4C2211yescharacteristic
15#5218—21no—
16C2×#832—21no—
17C2×#830C2211yescharacteristic
18C2×#830C2211yescharacteristic
19#5041—22no—
20#5040—21no—
21#5034—21no—
22#5033—22no—
23#5033—22no—
24#4971—21no—
25C2×#807—21no—
26#4943—22no—
27#4936—21no—
28#4929—21no—
29#4659—22no—
30#4529—22no—
31#4452—22no—
32#3187—31no—
33C2×#480—31no—
34#2503—31no—
35#1964D311yescharacteristic
36#1823—31no—
37#1520—31no—
38#1282—31no—
39C22×C3×D4—21no—
40C22×C3×D4D411yescharacteristic
41C23×C3×C4C2311yescharacteristic
42C23×Dic3—21no—
43C22×Dic6D411yescharacteristic
44C3×#121—21no—
45C3×#121—21no—
46C3×#121—21no—
47C3×#121—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4D411yescharacteristic
50C2×C3×C22⋊C4—21no—
51C2×C3×C22⋊C4—21no—
52C2×C3×C22⋊C4D412yes—
53C2×C3×C42—21no—
54C2×#217—21no—
55C2×#214—22no—
56C2×C4⋊Dic3—21no—
57C2×C4⋊Dic3—21no—
58C2×C4⋊Dic3—21no—
59C2×C4⋊Dic3—21no—
60C2×C4⋊Dic3—21no—
61C2×C4⋊Dic3D412yes—
62C2×C4⋊Dic3D411yescharacteristic
63#924—21no—
64#924—21no—
65#924—21no—
66#924—21no—
67C2×Dic3⋊C4—21no—
68C22×C3⋊C8—21no—
69C22×C3⋊C8D412yes—
70C3×#99—44no—
71C3×#95—21no—
72C3×#95—21no—
73#832—24no—
74#832—22no—
75C3⋊#95—24no—
76C3⋊#95—24no—
77#830—21no—
78#830—21no—
79#830—21no—
80#830D412yes—
81#810—22no—
82#808—22no—
83#807—24no—
84C2×C22⋊Q8—31no—
85C2×#121C2×D311yescharacteristic
86C22×C4⋊C4—31no—
87#483—61no—
88#481—61no—
89#480—62no—
90#480—62no—
91#465—62no—
92#450—61no—
93#448—61no—
94C2×#107—61no—
95C2×#102—61no—
96C2×#99—22no—
97C2×C22⋊C8—31no—
98C2×C22⋊C8—31no—
99#400—62no—
100#391—61no—
101#382—61no—
102#381—62no—
103#380C2×D311yescharacteristic
104C4×C22⋊C4C2×D311yescharacteristic
105#340—62no—
106#328—62no—
107C24×C3—21no—
108C24×C3C2×D411yescharacteristic
109C2×C3×D4—21no—
110C2×C3×D4—21no—
111C2×C3×D4—42no—
112C2×C3×D4—41no—
113C2×C3×D4—41no—
114C2×C3×D4—42no—
115C2×C3×D4—42no—
116C22×C3×C4—22no—
117C22×C3×C4—21no—
118C22×C3×C4—42no—
119C22×C3×C4—41no—
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4#4111yescharacteristic
126C22×Dic3—21no—
127C22×Dic3—21no—
128C22×Dic3—22no—
129C22×Dic3—41no—
130C22×Dic3—41no—
131C22×Dic3—42no—
132C22×Dic3—42no—
133C2×Dic6—21no—
134C2×Dic6—21no—
135C2×Dic6—42no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—42no—
139C3×C22⋊C4—22no—
140C3×C22⋊C4—22no—
141C3×C22⋊C4—41no—
142C3×C22⋊C4—41no—
143C3×C22⋊C4—42no—
144C3×C42—42no—
145#217—41no—
146#217—41no—
147#214—82no—
148C4⋊Dic3—22no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151C4⋊Dic3—22no—
152C4⋊Dic3—41no—
153C4⋊Dic3—41no—
154Dic3⋊C4—41no—
155Dic3⋊C4—41no—
156C2×C3⋊C8—22no—
157C2×C3⋊C8—22no—
158C2×C3⋊C8—42no—
159C2×C3⋊C8—42no—
160C22×Q8—31no—
161C22×D4—21no—
162C22×D4D1211yescharacteristic
163C23×C4—61no—
164C23×C4C22×D311yescharacteristic
165C2×SD16—62no—
166C22×C8—32no—
167C22×C8—61no—
168C22⋊Q8—61no—
169C22⋊Q8—61no—
170C22⋊Q8—61no—
171C22⋊Q8—61no—
172#121—21no—
173#121—21no—
174#121—21no—
175#121—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—32no—
178C2×C4⋊C4—31no—
179C2×C4⋊C4—61no—
180C2×C4⋊C4—61no—
181C2×C4⋊C4—61no—
182C2×C4⋊C4—61no—
183C2×C4⋊C4—61no—
184C2×C4⋊C4—61no—
185C2×C4⋊C4D1211yescharacteristic
186C2×C22⋊C4—21no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—61no—
189C2×C22⋊C4#6712yes—
190C2×C42—21no—
191#107—64no—
192#106—62no—
193#103—62no—
194#102—64no—
195#102—62no—
196#99—44no—
197C22⋊C8—32no—
198C22⋊C8—61no—
199C22⋊C8—61no—
200C22⋊C8—61no—
201#95—21no—
202#95—21no—
203#95—64no—
204#95—64no—
205C23×C3—41no—
206C23×C3—41no—
207C23×C3—41no—
208C23×C3—42no—
209C23×C3—81no—
210C23×C3—81no—
211C23×C3#12011yescharacteristic
212C23×C3#12111yescharacteristic
213C23×C3#12711yescharacteristic
214C23×C3#13512yes—
215C23×C3AGL(1,ℤ/8)12yes—
216C3×D4—42no—
217C3×D4—42no—
218C3×D4—82no—
219C3×D4—81no—
220C3×D4—81no—
221C2×C3×C4—22no—
222C2×C3×C4—22no—
223C2×C3×C4—22no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—42no—
228C2×C3×C4—42no—
229C2×C3×C4—41no—
230C2×C3×C4—42no—
231C2×C3×C4—42no—
232C2×C3×C4—81no—
233C2×C3×C4—81no—
234C2×C3×C4—81no—
235C2×C3×C4#12011yescharacteristic
236C2×C3×C4#12011yescharacteristic
237C2×C3×C4#12111yescharacteristic
238C2×C3×C4#12111yescharacteristic
239C2×Dic3—42no—
240C2×Dic3—42no—
241C2×Dic3—41no—
242C2×Dic3—41no—
243C2×Dic3—42no—
244C2×Dic3—44no—
245C2×Dic3—82no—
246C2×Dic3—81no—
247C2×Dic3—81no—
248Dic6—42no—
249Dic6—81no—
250C3⋊C8—42no—
251C3⋊C8—42no—
252C24—21no—
253C24C2×#6711yescharacteristic
254C2×Q8—61no—
255C2×Q8—61no—
256C2×Q8—122no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—42no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—42no—
267C22×C4—41no—
268C22×C4—61no—
269C22×C4—61no—
270C22×C4—62no—
271C22×C4—121no—
272C22×C4—122no—
273C22×C4—121no—
274C22×C4—122no—
275C22×C4C2×D1211yescharacteristic
276C22×C4#23511yescharacteristic
277C22×C4D3×D411yescharacteristic
278C22×C4D3×D411yescharacteristic
279C22×C4D3×D411yescharacteristic
280C22×C4D3×D411yescharacteristic
281SD16—242no—
282C2×C8—62no—
283C2×C8—62no—
284C2×C8—122no—
285C2×C8—122no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—42no—
289C4⋊C4—61no—
290C4⋊C4—62no—
291C4⋊C4—62no—
292C4⋊C4—61no—
293C4⋊C4—121no—
294C4⋊C4—121no—
295C4⋊C4—121no—
296C4⋊C4—121no—
297C22⋊C4—22no—
298C22⋊C4—22no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—42no—
302C22⋊C4—121no—
303C22⋊C4—121no—
304C42—42no—
305C22×C3—21no—
306C22×C3—21no—
307C22×C3—21no—
308C22×C3—21no—
309C22×C3—21no—
310C22×C3—21no—
311C22×C3—42no—
312C22×C3—42no—
313C22×C3—81no—
314C22×C3—81no—
315C22×C3—81no—
316C22×C3—162no—
317C22×C3#39211yescharacteristic
318C22×C3#44912yes—
319C22×C3#46312yes—
320C22×C3#49511yescharacteristic
321C22×C3#49611yescharacteristic
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—82no—
326C3×C4—81no—
327C3×C4—161no—
328C3×C4#45812yes—
329Dic3—81no—
330Dic3—82no—
331Dic3—81no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—42no—
336C23—81no—
337C23—81no—
338C23#93011yescharacteristic
339C23#93212yes—
340C23#93711yescharacteristic
341C23#94011yescharacteristic
342C23#95012yes—
343Q8—122no—
344Q8—241no—
345D4—42no—
346D4—42no—
347D4—82no—
348D4—81no—
349D4—81no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—122no—
365C2×C4—122no—
366C2×C4—121no—
367C2×C4—121no—
368C2×C4—122no—
369C2×C4—124no—
370C2×C4—242no—
371C2×C4—241no—
372C2×C4—241no—
373C2×C4#88211yescharacteristic
374C2×C4#88211yescharacteristic
375C2×C4#88411yescharacteristic
376C2×C4#88411yescharacteristic
377C8—122no—
378C8—122no—
379C2×C3—22no—
380C2×C3—21no—
381C2×C3—22no—
382C2×C3—21no—
383C2×C3—161no—
384C2×C3#200911yescharacteristic
385C2×C3#201211yescharacteristic
386C2×C3#202211yescharacteristic
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—42no—
394C22—42no—
395C22—81no—
396C22—81no—
397C22—81no—
398C22—162no—
399C22#496011yescharacteristic
400C22#501612yes—
401C22#503712yes—
402C22#505911yescharacteristic
403C22#507411yescharacteristic
404C4—21no—
405C4—21no—
406C4—21no—
407C4—82no—
408C4—81no—
409C4—161no—
410C4—242no—
411C4—241no—
412C4—241no—
413C4#474212yes—
414C3#995311yescharacteristic, complemented, Sylow
415C2—22no—
416C2—21no—
417C2—22no—
418C2—21no—
419C2—161no—
420C2#6965911yescharacteristic
421C2#7003111yescharacteristic
422C2#7005311yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1964C2
socleC22×C3#392
prime radicalC2×C3×#121C22
hypercenter#1964D3
hypocenterC3#9953
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1964

Complements

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

NH ≅ G/Ntype
3. #73518C2semidirect
5. #69510C2semidirect
6. #69501C2semidirect
7. #67617C2semidirect
414. C3#9953semidirect

Maximal subgroups

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

subgroupindex
#826302
C2×#50332
#735182
C3×#19642
#695102
#695012
#676172
#99533