Finite Group Databasebeta

← #10692349

Subgroups — #10692349

Order 768.

760 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 760 conjugacy classes of subgroups — too many to draw. Only its 97 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#78640C211yesmaximal, characteristic
2#73573C211yesmaximal, characteristic
3#69457C211yesmaximal, characteristic, complemented
4#69454C211yesmaximal, characteristic
5#69445C211yesmaximal, characteristic, complemented
6C3×#1297C211yesmaximal, characteristic, complemented
7#67684C211yesmaximal, characteristic, complemented
8#9361—31nomaximal, Sylow
9C22×#235C2211yescharacteristic
10C2×#924C2211yescharacteristic
11C2×#909C412yescomplemented
12C2×C4×D12—21no—
13C3×#407C2211yescharacteristic
14C3×C4×C22⋊C4C2211yescharacteristic
15C3×C4×C22⋊C4C2211yescharacteristic
16#5212C2211yescharacteristic
17#5207C2211yescharacteristic
18#5133C412yescomplemented
19#4956—22no—
20#4929—21no—
21C2×#805—22no—
22#4739—21no—
23#4733—22no—
24#4731—21no—
25#4664—21no—
26D3×#95—21no—
27#4527—22no—
28#4452—21no—
29#4452—21no—
30#3007—31no—
31#2339—31no—
32#1996—31no—
33#1789—31no—
34#1508—31no—
35#1305—31no—
36#1297D311yescharacteristic, complemented
37C23×C3×C4C2311yescharacteristic
38C22×#67—21no—
39C2×#235—21no—
40C2×#235—21no—
41C2×#235—44no—
42C2×#235C2×C412yes—
43C2×#235D412yes—
44C22×D12D411yescharacteristic
45C22×C4×D3—21no—
46C22×C4×D3—22no—
47C22×Dic6C2×C411yescharacteristic, complemented
48C22×Dic6D411yescharacteristic
49C2×C3×M16C2×C412yescomplemented
50C2×C3×C22⋊C4—21no—
51C2×C3×C22⋊C4—21no—
52C2×C3×C42—21no—
53C2×C3×C42—21no—
54C2×#217—21no—
55C2×#217D412yes—
56C2×(C2×D3)⋊C4—22no—
57C2×C4⋊Dic3—21no—
58C2×C4⋊Dic3C2×C411yescharacteristic, complemented
59#924—21no—
60#924—21no—
61#924—21no—
62#924—21no—
63C2×Dic3⋊C4—21no—
64C2×#208—21no—
65#909—44no—
66C2×Dic12—22no—
67C2×#204—22no—
68C4×D12—44no—
69C3×C22⋊C8—21no—
70C3×C22⋊C8—21no—
71C3×#95—21no—
72C3×#95D411yescharacteristic
73C3×#95D411yescharacteristic
74C3⋊#95—22no—
75C3⋊#95—22no—
76C3⋊#95—21no—
77#830—21no—
78#830—21no—
79#821—22no—
80#816—22no—
81#805—44no—
82C22×#41—31no—
83C2×#137—32no—
84C2×C22⋊Q8—31no—
85C2×C4×D4—61no—
86#452—62no—
87#451—61no—
88#450—61no—
89C2×#104—62no—
90#417—32no—
91#407—31no—
92#407C2×D311yescharacteristic
93#391—61no—
94#386—62no—
95#381—61no—
96#380—31no—
97C4×C22⋊C4C2×D311yescharacteristic
98C4×C22⋊C4C2×D311yescharacteristic
99C2×#95—61no—
100#342—62no—
101#328—61no—
102#328—61no—
103C24×C3C2×D411yescharacteristic
104C23×D3—21no—
105C22×C3×C4—21no—
106C22×C3×C4—21no—
107C22×C3×C4—42no—
108C22×C3×C4—41no—
109C22×C3×C4C22×C411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4#4111yescharacteristic
114C22×C3×C4#4111yescharacteristic
115C2×#67—22no—
116C2×#67—22no—
117C2×#67—44no—
118C22×Dic3—21no—
119C22×Dic3—22no—
120C22×Dic3—21no—
121C22×Dic3—21no—
122#235—24no—
123#235—24no—
124#235—42no—
125#235—42no—
126#235—44no—
127#235—44no—
128C2×D12—21no—
129C2×D12—24no—
130C2×D12C22⋊C412yes—
131C2×C4×D3—22no—
132C2×C4×D3—22no—
133C2×C4×D3—44no—
134C2×C4×D3—44no—
135C2×C4×D3—44no—
136C2×C4×D3—82no—
137C2×Dic6—21no—
138C2×Dic6—21no—
139C2×Dic6—21no—
140C2×Dic6—42no—
141C2×Dic6—42no—
142C2×Dic6C22⋊C412yes—
143C3×M16—42no—
144C2×C3×C8—22no—
145C3×C22⋊C4—22no—
146C3×C22⋊C4—42no—
147C3×C22⋊C4—41no—
148C3×C42—42no—
149C3×C42—42no—
150#217—41no—
151#217—41no—
152#217—42no—
153(C2×D3)⋊C4—44no—
154C4⋊Dic3—21no—
155C4⋊Dic3—21no—
156C4⋊Dic3—42no—
157Dic3⋊C4—41no—
158Dic3⋊C4—41no—
159#208—42no—
160C2×C3⋊C8—41no—
161Dic12—44no—
162#204—44no—
163C2×#41—32no—
164C2×#41—32no—
165C2×#41—61no—
166C2×#41—61no—
167C2×#41—124no—
168C22×Q8—31no—
169C22×Q8—31no—
170C22×D4—31no—
171C22×D4—61no—
172C23×C4—62no—
173C23×C4—61no—
174C23×C4C22×D311yescharacteristic
175#137—124no—
176C2×Q16—62no—
177C2×SD16—62no—
178C2×M16—61no—
179C2×M16C4×D312yescomplemented
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—21no—
190C2×C22⋊C4—32no—
191C2×C22⋊C4—62no—
192C2×C22⋊C4—61no—
193C2×C42—21no—
194C2×C42—21no—
195#104—124no—
196#103—62no—
197#102—62no—
198C22⋊C8—21no—
199C22⋊C8—21no—
200C22⋊C8—61no—
201C22⋊C8—61no—
202#95—21no—
203#95—62no—
204#95—62no—
205#95—61no—
206#95D1211yescharacteristic
207#95D1211yescharacteristic
208C23×C3—21no—
209C23×C3—21no—
210C23×C3—41no—
211C23×C3—41no—
212C23×C3C4×D411yescharacteristic
213C23×C3#12011yescharacteristic
214C23×C3#12111yescharacteristic
215C22×D3—24no—
216C22×D3—22no—
217C22×D3—42no—
218C22×D3—41no—
219C22×D3—41no—
220C2×C3×C4—22no—
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—41no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—42no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—81no—
234C2×C3×C4—81no—
235C2×C3×C4C2×C22⋊C411yescharacteristic
236C2×C3×C4C4×D411yescharacteristic
237C2×C3×C4#12111yescharacteristic
238C2×C3×C4#12311yescharacteristic
239#67—44no—
240#67—44no—
241C2×Dic3—22no—
242C2×Dic3—42no—
243C2×Dic3—42no—
244C2×Dic3—42no—
245C2×Dic3—41no—
246C2×Dic3—41no—
247C2×Dic3—42no—
248C2×Dic3—81no—
249C2×Dic3—81no—
250D12—24no—
251D12—22no—
252D12#9914yes—
253C4×D3—44no—
254C4×D3—44no—
255C4×D3—88no—
256Dic6—22no—
257Dic6—22no—
258Dic6—42no—
259Dic6—42no—
260Dic6—81no—
261C3×C8—42no—
262C3⋊C8—81no—
263C24—61no—
264C24D3×D411yescharacteristic
265#41—64no—
266#41—64no—
267#41—122no—
268#41—122no—
269#41—124no—
270#41—124no—
271C2×Q8—32no—
272C2×Q8—61no—
273C2×Q8—61no—
274C2×Q8—61no—
275C2×Q8—122no—
276C2×Q8—122no—
277C2×D4—32no—
278C2×D4—64no—
279C2×D4—62no—
280C2×D4—62no—
281C2×D4—61no—
282C2×D4—124no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4—61no—
288C22×C4—62no—
289C22×C4—62no—
290C22×C4—62no—
291C22×C4—61no—
292C22×C4—61no—
293C22×C4—124no—
294C22×C4—124no—
295C22×C4—124no—
296C22×C4—242no—
297C22×C4C2×C4×D311yescharacteristic
298C22×C4C2×D1211yescharacteristic
299C22×C4#23511yescharacteristic
300C22×C4D3×D411yescharacteristic
301C22×C4D3×D411yescharacteristic
302C22×C4#23911yescharacteristic
303Q16—124no—
304SD16—124no—
305M16—42no—
306M16—122no—
307C2×C8—22no—
308C2×C8—121no—
309C4⋊C4—61no—
310C4⋊C4—61no—
311C4⋊C4—122no—
312C4⋊C4—121no—
313C4⋊C4—121no—
314C22⋊C4—22no—
315C22⋊C4—42no—
316C22⋊C4—41no—
317C22⋊C4—124no—
318C22⋊C4—121no—
319C22⋊C4—121no—
320C22⋊C4—122no—
321C42—42no—
322C42—42no—
323C22×C3—21no—
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—21no—
327C22×C3—21no—
328C22×C3—21no—
329C22×C3—21no—
330C22×C3—21no—
331C22×C3—81no—
332C22×C3—81no—
333C22×C3#38611yescharacteristic
334C22×C3#45511yescharacteristic
335C22×C3#45611yescharacteristic
336C2×D3—24no—
337C2×D3—44no—
338C2×D3—44no—
339C2×D3—42no—
340C2×D3—42no—
341C3×C4—21no—
342C3×C4—21no—
343C3×C4—21no—
344C3×C4—42no—
345C3×C4—81no—
346C3×C4—82no—
347C3×C4C2×#9912yes—
348Dic3—42no—
349Dic3—82no—
350Dic3—81no—
351Dic3—81no—
352C23—21no—
353C23—21no—
354C23—41no—
355C23—41no—
356C23—64no—
357C23—62no—
358C23—122no—
359C23—121no—
360C23—121no—
361C23#88211yescharacteristic
362C23#88411yescharacteristic
363C23#89311yescharacteristic
364Q8—62no—
365Q8—62no—
366Q8—122no—
367Q8—122no—
368Q8—241no—
369D4—34no—
370D4—64no—
371D4—62no—
372D4—124no—
373D4—124no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—62no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—124no—
391C2×C4—124no—
392C2×C4—122no—
393C2×C4—122no—
394C2×C4—122no—
395C2×C4—121no—
396C2×C4—121no—
397C2×C4—122no—
398C2×C4—248no—
399C2×C4—241no—
400C2×C4—241no—
401C2×C4C4×D1211yescharacteristic
402C2×C4D3×C22⋊C411yescharacteristic
403C2×C4#89411yescharacteristic
404C2×C4#89511yescharacteristic
405C8—42no—
406C8—241no—
407C2×C3—21no—
408C2×C3—21no—
409C2×C3—41no—
410C2×C3—41no—
411C2×C3#188211yescharacteristic
412C2×C3#189811yescharacteristic
413C2×C3#190611yescharacteristic
414D3—44no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—64no—
424C22—81no—
425C22—81no—
426C22—124no—
427C22—124no—
428C22—122no—
429C22—122no—
430C22#466511yescharacteristic
431C22#482511yescharacteristic
432C22#482611yescharacteristic
433C4—21no—
434C4—21no—
435C4—21no—
436C4—42no—
437C4—81no—
438C4—82no—
439C4—122no—
440C4—242no—
441C4—241no—
442C4—241no—
443C4D3×#9912yes—
444C3#936111yescharacteristic, complemented, Sylow
445C2—21no—
446C2—21no—
447C2—41no—
448C2—41no—
449C2—124no—
450C2#6877511yescharacteristic
451C2#6877811yescharacteristic
452C2#6897011yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1297C2
socleC22×C3#386
prime radicalC22×#235C22
cosocleC23×C3×C4C23
hypercenter#1297D3
hypocenterC3#9361
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1297

Complements

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

NH ≅ G/Ntype
3. #69457C2semidirect
5. #69445C2semidirect
6. C3×#1297C2semidirect
7. #67684C2semidirect
11. C2×#909C4semidirect
18. #5133C4semidirect
36. #1297D3semidirect
47. C22×Dic6C2×C4semidirect
49. C2×C3×M16C2×C4semidirect
58. C2×C4⋊Dic3C2×C4semidirect
179. C2×M16C4×D3semidirect
444. C3#9361semidirect

Maximal subgroups

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

subgroupindex
#786402
#735732
#694572
#694542
#694452
C3×#12972
#676842
#93613