Finite Group Databasebeta

← #10692346

Subgroups — #10692346

Order 768.

760 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 760 conjugacy classes of subgroups — too many to draw. Only its 107 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#84088C211yesmaximal, characteristic
2#84084C211yesmaximal, characteristic
3C3×#1789C211yesmaximal, characteristic
4#69980C211yesmaximal, characteristic, complemented
5#69457C211yesmaximal, characteristic, complemented
6#67685C211yesmaximal, characteristic, complemented
7#67618C211yesmaximal, characteristic, complemented
8#9361—31nomaximal, Sylow
9C22×C3×#41C2211yescharacteristic
10C2×#951C412yescomplemented
11C2×D4×Dic3—21no—
12C2×#924C2211yescharacteristic
13C2×C3×#104—22no—
14C3×#407C2211yescharacteristic
15C3×C4×C22⋊C4C2211yescharacteristic
16#5239—22no—
17#5232C412yescomplemented
18#5229—21no—
19#5228—21no—
20#5220—22no—
21#5218—21no—
22#5212C2211yescharacteristic
23#5211—21no—
24C4×#217C2211yescharacteristic
25C2×C3⋊#95—21no—
26#5207C2211yescharacteristic
27#4527—22no—
28#4467—21no—
29#4467—21no—
30#3007—31no—
31#2339—31no—
32#1996—31no—
33#1789D311yescharacteristic
34#1508—31no—
35#1305—31no—
36#1297—31no—
37C2×C3×#41—21no—
38C2×C3×#41—21no—
39C2×C3×#41—44no—
40C2×C3×#41C2×C412yes—
41C2×C3×#41D412yes—
42C22×C3×Q8D411yescharacteristic
43C22×C3×D4—21no—
44C22×C3×D4D411yescharacteristic
45C23×C3×C4—21no—
46C23×C3×C4C2311yescharacteristic
47C23×Dic3—22no—
48C22×Dic6C2×C411yescharacteristic, complemented
49C2×C3×M16—21no—
50C2×C3×C22⋊C4—21no—
51C2×C3×C42—21no—
52C2×#217—22no—
53C2×#217—21no—
54C2×#217—21no—
55C2×#217D412yes—
56#951—44no—
57C2×#216—22no—
58D4×Dic3—44no—
59C2×#214—22no—
60C2×C4⋊Dic3—21no—
61C2×C4⋊Dic3C2×C411yescharacteristic, complemented
62#924—21no—
63#924—21no—
64#924—21no—
65#924—21no—
66C2×Dic3⋊C4—21no—
67C2×C4×Dic3—21no—
68C2×#208C2×C412yescomplemented
69C3×#104—44no—
70C3×C22⋊C8—21no—
71C3×C22⋊C8—21no—
72C3×#95—21no—
73#835—22no—
74#832—22no—
75C3⋊#95—22no—
76C3⋊#95—22no—
77C3⋊#95—21no—
78C3⋊#95D411yescharacteristic
79C3⋊#95D411yescharacteristic
80#830—21no—
81#830—21no—
82C22×#41C2×D311yescharacteristic
83C2×#137—32no—
84C2×C22⋊Q8—31no—
85C2×C4×D4—61no—
86#452—62no—
87#451—61no—
88#450—61no—
89C2×#104—22no—
90#417—32no—
91#407—31no—
92#407C2×D311yescharacteristic
93#391—61no—
94#386—62no—
95#381—61no—
96#380—31no—
97C4×C22⋊C4—31no—
98C4×C22⋊C4C2×D311yescharacteristic
99C2×#95—61no—
100#342—62no—
101#328—61no—
102#328—61no—
103C24×C3—21no—
104C24×C3C2×D411yescharacteristic
105C3×#41—24no—
106C3×#41—24no—
107C3×#41—42no—
108C3×#41—42no—
109C3×#41—44no—
110C3×#41—44no—
111C2×C3×Q8—21no—
112C2×C3×Q8—42no—
113C2×C3×Q8C22⋊C412yes—
114C2×C3×D4—24no—
115C2×C3×D4—22no—
116C2×C3×D4—22no—
117C2×C3×D4—21no—
118C2×C3×D4—44no—
119C2×C3×D4C22⋊C412yes—
120C22×C3×C4—21no—
121C22×C3×C4—22no—
122C22×C3×C4—22no—
123C22×C3×C4—42no—
124C22×C3×C4—44no—
125C22×C3×C4—41no—
126C22×C3×C4C22×C411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4#4111yescharacteristic
131C22×C3×C4#4111yescharacteristic
132C22×Dic3—21no—
133C22×Dic3—22no—
134C22×Dic3—21no—
135C22×Dic3—21no—
136C22×Dic3—21no—
137C22×Dic3—44no—
138C22×Dic3—44no—
139C22×Dic3—82no—
140C2×Dic6—21no—
141C2×Dic6—21no—
142C2×Dic6—42no—
143C3×M16—42no—
144C2×C3×C8—41no—
145C3×C22⋊C4—22no—
146C3×C22⋊C4—41no—
147C3×C42—42no—
148#217—42no—
149#217—44no—
150#217—41no—
151#217—41no—
152#217—42no—
153#216—44no—
154#214—44no—
155C4⋊Dic3—21no—
156C4⋊Dic3—21no—
157C4⋊Dic3—42no—
158Dic3⋊C4—41no—
159Dic3⋊C4—41no—
160C4×Dic3—42no—
161#208—42no—
162C2×C3⋊C8—22no—
163C2×#41—21no—
164C2×#41—21no—
165C2×#41—44no—
166C2×#41C4×D312yes—
167C2×#41#6712yes—
168C22×Q8—31no—
169C22×Q8D1211yescharacteristic
170C22×D4—21no—
171C22×D4D1211yescharacteristic
172C23×C4—21no—
173C23×C4—62no—
174C23×C4C22×D311yescharacteristic
175#137—124no—
176C2×Q16—62no—
177C2×SD16—62no—
178C2×M16—21no—
179C2×M16—32no—
180C22⋊Q8—61no—
181C22⋊Q8—61no—
182C22⋊Q8—61no—
183C22⋊Q8—61no—
184C4×D4—124no—
185C2×C4⋊C4—31no—
186C2×C4⋊C4—61no—
187C2×C4⋊C4—61no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—32no—
190C2×C22⋊C4—62no—
191C2×C22⋊C4—61no—
192C2×C22⋊C4—61no—
193C2×C42—21no—
194C2×C42—61no—
195#104—44no—
196#103—62no—
197#102—62no—
198C22⋊C8—21no—
199C22⋊C8—21no—
200C22⋊C8—61no—
201C22⋊C8—61no—
202#95—21no—
203#95—31no—
204#95—31no—
205#95—62no—
206#95—62no—
207#95—61no—
208C23×C3—24no—
209C23×C3—22no—
210C23×C3—21no—
211C23×C3—21no—
212C23×C3—41no—
213C23×C3—41no—
214C23×C3—42no—
215C23×C3—41no—
216C23×C3—41no—
217C23×C3C4×D411yescharacteristic
218C23×C3#12011yescharacteristic
219C23×C3#12111yescharacteristic
220C3×Q8—22no—
221C3×Q8—22no—
222C3×Q8—42no—
223C3×D4—24no—
224C3×D4—22no—
225C3×D4—44no—
226C3×D4—44no—
227C3×D4#9914yes—
228C2×C3×C4—22no—
229C2×C3×C4—22no—
230C2×C3×C4—22no—
231C2×C3×C4—22no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—44no—
236C2×C3×C4—44no—
237C2×C3×C4—42no—
238C2×C3×C4—42no—
239C2×C3×C4—81no—
240C2×C3×C4—81no—
241C2×C3×C4C2×C22⋊C411yescharacteristic
242C2×C3×C4C4×D411yescharacteristic
243C2×C3×C4#12111yescharacteristic
244C2×C3×C4#12311yescharacteristic
245C2×Dic3—22no—
246C2×Dic3—41no—
247C2×Dic3—41no—
248C2×Dic3—41no—
249C2×Dic3—42no—
250C2×Dic3—41no—
251C2×Dic3—42no—
252C2×Dic3—41no—
253C2×Dic3—41no—
254C2×Dic3—42no—
255C2×Dic3—41no—
256C2×Dic3—88no—
257C2×Dic3—81no—
258C2×Dic3—81no—
259Dic6—42no—
260Dic6—81no—
261C3×C8—81no—
262C3⋊C8—42no—
263C24—21no—
264C24D3×D411yescharacteristic
265#41—24no—
266#41—24no—
267#41—42no—
268#41—42no—
269#41—44no—
270#41—44no—
271C2×Q8—21no—
272C2×Q8—42no—
273C2×Q8—61no—
274C2×Q8—61no—
275C2×Q8—122no—
276C2×Q8(C2×D3)⋊C412yes—
277C2×D4—24no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—21no—
281C2×D4—44no—
282C2×D4(C2×D3)⋊C412yes—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—42no—
287C22×C4—44no—
288C22×C4—41no—
289C22×C4—61no—
290C22×C4—61no—
291C22×C4—62no—
292C22×C4—61no—
293C22×C4—61no—
294C22×C4—124no—
295C22×C4—124no—
296C22×C4—242no—
297C22×C4C2×C4×D311yescharacteristic
298C22×C4C2×D1211yescharacteristic
299C22×C4D3×D411yescharacteristic
300C22×C4#23711yescharacteristic
301C22×C4#23711yescharacteristic
302C22×C4C2×#6711yescharacteristic
303Q16—124no—
304SD16—124no—
305M16—42no—
306M16—122no—
307C2×C8—41no—
308C2×C8—62no—
309C4⋊C4—61no—
310C4⋊C4—61no—
311C4⋊C4—122no—
312C4⋊C4—121no—
313C4⋊C4—121no—
314C22⋊C4—22no—
315C22⋊C4—41no—
316C22⋊C4—122no—
317C22⋊C4—124no—
318C22⋊C4—121no—
319C22⋊C4—121no—
320C22⋊C4—122no—
321C42—42no—
322C42—122no—
323C22×C3—21no—
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—21no—
327C22×C3—21no—
328C22×C3—21no—
329C22×C3—24no—
330C22×C3—21no—
331C22×C3—21no—
332C22×C3—44no—
333C22×C3—44no—
334C22×C3—42no—
335C22×C3—42no—
336C22×C3—81no—
337C22×C3—81no—
338C22×C3#38611yescharacteristic
339C22×C3#45511yescharacteristic
340C22×C3#45611yescharacteristic
341C3×C4—21no—
342C3×C4—21no—
343C3×C4—21no—
344C3×C4—42no—
345C3×C4—82no—
346C3×C4C2×#9912yes—
347Dic3—42no—
348Dic3—81no—
349Dic3—82no—
350Dic3—81no—
351Dic3—81no—
352C23—24no—
353C23—22no—
354C23—21no—
355C23—21no—
356C23—41no—
357C23—41no—
358C23—42no—
359C23—41no—
360C23—41no—
361C23#67⋊C411yescharacteristic
362C23#88211yescharacteristic
363C23#93911yescharacteristic
364Q8—22no—
365Q8—22no—
366Q8—42no—
367Q8—122no—
368Q8—241no—
369D4—24no—
370D4—22no—
371D4—44no—
372D4—44no—
373D4#80614yes—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—62no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—121no—
389C2×C4—121no—
390C2×C4—121no—
391C2×C4—122no—
392C2×C4—121no—
393C2×C4—122no—
394C2×C4—121no—
395C2×C4—121no—
396C2×C4—122no—
397C2×C4—121no—
398C2×C4—248no—
399C2×C4—241no—
400C2×C4—241no—
401C2×C4#67⋊C411yescharacteristic
402C2×C4#88611yescharacteristic
403C2×C4C2×(C2×D3)⋊C411yescharacteristic
404C2×C4#93911yescharacteristic
405C8—81no—
406C8—122no—
407C2×C3—21no—
408C2×C3—21no—
409C2×C3—41no—
410C2×C3—41no—
411C2×C3—44no—
412C2×C3#188211yescharacteristic
413C2×C3#189811yescharacteristic
414C2×C3#190611yescharacteristic
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—24no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—44no—
425C22—44no—
426C22—42no—
427C22—42no—
428C22—81no—
429C22—81no—
430C22#482511yescharacteristic
431C22#482611yescharacteristic
432C22#495811yescharacteristic
433C4—21no—
434C4—21no—
435C4—21no—
436C4—42no—
437C4—82no—
438C4—122no—
439C4—242no—
440C4—241no—
441C4—241no—
442C4—241no—
443C4C2×#80612yes—
444C3#936111yescharacteristic, complemented, Sylow
445C2—21no—
446C2—21no—
447C2—41no—
448C2—41no—
449C2—44no—
450C2#6949611yescharacteristic
451C2#7036511yescharacteristic
452C2#7039511yescharacteristic
subgroupG/H
centerC22#4958
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1789C2
socleC22×C3#386
prime radicalC22×C3×#41C22
cosocleC23×C3×C4C23
hypercenter#1789D3
hypocenterC3#9361
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1789

Complements

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

NH ≅ G/Ntype
4. #69980C2semidirect
5. #69457C2semidirect
6. #67685C2semidirect
7. #67618C2semidirect
10. C2×#951C4semidirect
17. #5232C4semidirect
48. C22×Dic6C2×C4semidirect
61. C2×C4⋊Dic3C2×C4semidirect
68. C2×#208C2×C4semidirect
444. C3#9361semidirect

Maximal subgroups

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

subgroupindex
#840882
#840842
C3×#17892
#699802
#694572
#676852
#676182
#93613