Finite Group Databasebeta

← #10692371

Subgroups — #10692371

Order 768.

732 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 115 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
1C3×#3007C211yesmaximal, characteristic
2#84089C211yesmaximal, characteristic
3#72138C211yesmaximal, characteristic
4#71701C211yesmaximal, characteristic, complemented
5#71688C211yesmaximal, characteristic, complemented
6#67684C211yesmaximal, characteristic, complemented
7#67683C211yesmaximal, characteristic, complemented
8#9364—31nomaximal, Sylow
9C22×C3×#41C2211yescharacteristic
10C2×C3×#137C412yescomplemented
11C2×C3×C22⋊Q8C2211yescharacteristic
12C2×Q8×Dic3—21no—
13C3×#452—22no—
14C3×#451—21no—
15C3×#450—21no—
16C3×#417C412yescomplemented
17C3×#407C2211yescharacteristic
18C2×#837—22no—
19#5221—21no—
20#5221—22no—
21#5212—21no—
22#5212C2211yescharacteristic
23C4×#217—21no—
24C4×#217C2211yescharacteristic
25C4×#217C2211yescharacteristic
26#5207C2211yescharacteristic
27#4538—21no—
28#4538—21no—
29#4526—22no—
30#3007D311yescharacteristic
31#2341—31no—
32#1996—31no—
33#1789—31no—
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×Q8C2×C411yescharacteristic
43C22×C3×Q8D411yescharacteristic
44C22×C3×D4—21no—
45C22×C3×D4D411yescharacteristic
46C23×C3×C4—21no—
47C23×C3×C4C2311yescharacteristic
48C3×#137—44no—
49C2×C3×Q16—22no—
50C2×C3×SD16—22no—
51C2×C3×M16C2×C412yes—
52C3×C22⋊Q8—21no—
53C3×C22⋊Q8—21no—
54C3×C22⋊Q8—21no—
55C3×C22⋊Q8—21no—
56C2×C3×C4⋊C4—21no—
57C2×C3×C4⋊C4C2×C411yescharacteristic
58C2×C3×C22⋊C4—21no—
59C2×#217—21no—
60C2×#217—22no—
61C2×#217—21no—
62C2×#217—22no—
63C2×#217D412yes—
64Q8×Dic3—44no—
65C2×C4⋊Dic3—22no—
66C2×C4⋊Dic3—21no—
67C2×C4×Dic3—22no—
68C2×C4×Dic3—21no—
69C2×C4×Dic3—21no—
70C2×#208—21no—
71C3×#103—22no—
72C3×#102—22no—
73C3×C22⋊C8—21no—
74C3×C22⋊C8—21no—
75#837—44no—
76C3⋊#95—21no—
77C3⋊#95—21no—
78C3⋊#95D411yescharacteristic
79C3⋊#95D411yescharacteristic
80#830—21no—
81#830—21no—
82C22×#41C2×D311yescharacteristic
83C2×#137Dic312yescomplemented
84C2×C22⋊Q8C2×D311yescharacteristic
85C2×C4×Q8—61no—
86#452—22no—
87#451—21no—
88#450—21no—
89C2×#104—62no—
90#417Dic312yescomplemented
91#407—31no—
92#407C2×D311yescharacteristic
93#390—62no—
94#390—61no—
95#380—31no—
96#380—61no—
97C4×C22⋊C4—31no—
98C4×C22⋊C4—31no—
99C4×C22⋊C4—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—24no—
112C2×C3×Q8—21no—
113C2×C3×Q8—21no—
114C2×C3×Q8—21no—
115C2×C3×Q8—42no—
116C2×C3×Q8C22⋊C412yes—
117C2×C3×D4—22no—
118C2×C3×D4—22no—
119C2×C3×D4—21no—
120C2×C3×D4—42no—
121C2×C3×D4—44no—
122C2×C3×D4C22⋊C412yes—
123C22×C3×C4—21no—
124C22×C3×C4—22no—
125C22×C3×C4—22no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—42no—
129C22×C3×C4—44no—
130C22×C3×C4C22×C411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4#4111yescharacteristic
135C22×C3×C4#4111yescharacteristic
136C22×Dic3—21no—
137C22×Dic3—22no—
138C22×Dic3—21no—
139C22×Dic3—41no—
140C3×Q16—44no—
141C3×SD16—44no—
142C3×M16—42no—
143C2×C3×C8—22no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—41no—
147C3×C4⋊C4—41no—
148C3×C22⋊C4—41no—
149C3×C22⋊C4—41no—
150#217—22no—
151#217—42no—
152#217—44no—
153#217—41no—
154#217—42no—
155#217—82no—
156C4⋊Dic3—44no—
157C4⋊Dic3—42no—
158C4×Dic3—44no—
159C4×Dic3—42no—
160C4×Dic3—42no—
161#208—42no—
162C2×C3⋊C8—41no—
163C2×#41—21no—
164C2×#41—21no—
165C2×#41—44no—
166C2×#41C2×Dic312yes—
167C2×#41#6712yes—
168C22×Q8C2×Dic311yescharacteristic
169C22×Q8#6711yescharacteristic
170C22×D4—21no—
171C22×D4#6711yescharacteristic
172C23×C4—21no—
173C23×C4C22×D311yescharacteristic
174#137—44no—
175C2×Q16—22no—
176C2×SD16—22no—
177C2×M16—61no—
178C2×M16C2×Dic312yes—
179C22⋊Q8—21no—
180C22⋊Q8—21no—
181C22⋊Q8—21no—
182C22⋊Q8—21no—
183C4×Q8—124no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—62no—
186C2×C4⋊C4—61no—
187C2×C4⋊C4C2×Dic311yescharacteristic
188C2×C22⋊C4—21no—
189C2×C22⋊C4—32no—
190C2×C22⋊C4—61no—
191C2×C22⋊C4—62no—
192C2×C22⋊C4—61no—
193C2×C22⋊C4—62no—
194C2×C42—62no—
195C2×C42—61no—
196C2×C42—61no—
197#104—124no—
198#103—22no—
199#102—22no—
200C22⋊C8—21no—
201C22⋊C8—21no—
202C22⋊C8—61no—
203C22⋊C8—61no—
204#95—31no—
205#95—31no—
206#95—61no—
207#95—61no—
208C23×C3—21no—
209C23×C3—21no—
210C23×C3—22no—
211C23×C3—41no—
212C23×C3—41no—
213C23×C3—42no—
214C23×C3—42no—
215C23×C3—41no—
216C23×C3—41no—
217C23×C3C4×D411yescharacteristic
218C23×C3#12011yescharacteristic
219C23×C3#12111yescharacteristic
220C3×Q8—24no—
221C3×Q8—22no—
222C3×Q8—42no—
223C3×Q8—81no—
224C3×Q8#9914yes—
225C3×D4—22no—
226C3×D4—22no—
227C3×D4—44no—
228C3×D4—44no—
229C3×D4—42no—
230C2×C3×C4—24no—
231C2×C3×C4—22no—
232C2×C3×C4—22no—
233C2×C3×C4—22no—
234C2×C3×C4—22no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—44no—
239C2×C3×C4—44no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—81no—
243C2×C3×C4—81no—
244C2×C3×C4C2×C22⋊C411yescharacteristic
245C2×C3×C4C4×D411yescharacteristic
246C2×C3×C4#12111yescharacteristic
247C2×C3×C4#12311yescharacteristic
248C2×Dic3—22no—
249C2×Dic3—41no—
250C2×Dic3—41no—
251C2×Dic3—42no—
252C2×Dic3—41no—
253C2×Dic3—42no—
254C2×Dic3—42no—
255C2×Dic3—41no—
256C2×Dic3—42no—
257C2×Dic3—41no—
258C2×Dic3—81no—
259C2×Dic3—81no—
260C3×C8—42no—
261C3⋊C8—81no—
262C24—21no—
263C24D3×D411yescharacteristic
264#41—24no—
265#41—24no—
266#41—42no—
267#41—42no—
268#41—44no—
269#41—44no—
270C2×Q8—24no—
271C2×Q8—21no—
272C2×Q8—21no—
273C2×Q8—21no—
274C2×Q8—42no—
275C2×Q8#21712yes—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—21no—
279C2×D4—42no—
280C2×D4—44no—
281C2×D4#21712yes—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—42no—
288C22×C4—44no—
289C22×C4—61no—
290C22×C4—61no—
291C22×C4—62no—
292C22×C4—121no—
293C22×C4D3×D411yescharacteristic
294C22×C4#23711yescharacteristic
295C22×C4#23711yescharacteristic
296C22×C4C22×Dic311yescharacteristic
297C22×C4C2×#6711yescharacteristic
298C22×C4C2×#6711yescharacteristic
299Q16—44no—
300SD16—44no—
301M16—42no—
302M16—122no—
303C2×C8—22no—
304C2×C8—121no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—124no—
310C4⋊C4—122no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—62no—
314C22⋊C4—122no—
315C22⋊C4—124no—
316C22⋊C4—121no—
317C22⋊C4—122no—
318C22⋊C4—242no—
319C42—124no—
320C42—122no—
321C42—122no—
322C22×C3—21no—
323C22×C3—21no—
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—21no—
327C22×C3—21no—
328C22×C3—21no—
329C22×C3—21no—
330C22×C3—42no—
331C22×C3—42no—
332C22×C3—42no—
333C22×C3—81no—
334C22×C3—81no—
335C22×C3—82no—
336C22×C3—82no—
337C22×C3#38611yescharacteristic
338C22×C3#45511yescharacteristic
339C22×C3#45611yescharacteristic
340C3×C4—21no—
341C3×C4—24no—
342C3×C4—21no—
343C3×C4—21no—
344C3×C4—81no—
345C3×C4—81no—
346C3×C4C2×#9912yes—
347Dic3—42no—
348Dic3—82no—
349Dic3—81no—
350Dic3—82no—
351C23—21no—
352C23—21no—
353C23—22no—
354C23—41no—
355C23—41no—
356C23—42no—
357C23—42no—
358C23—41no—
359C23—41no—
360C23D4×Dic311yescharacteristic
361C23#93711yescharacteristic
362C23#93911yescharacteristic
363Q8—24no—
364Q8—22no—
365Q8—42no—
366Q8—81no—
367Q8#83414yes—
368D4—22no—
369D4—22no—
370D4—44no—
371D4—44no—
372D4—42no—
373C2×C4—24no—
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—41no—
384C2×C4—41no—
385C2×C4—62no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—121no—
389C2×C4—121no—
390C2×C4—121no—
391C2×C4—122no—
392C2×C4—122no—
393C2×C4—121no—
394C2×C4—122no—
395C2×C4—122no—
396C2×C4—121no—
397C2×C4—241no—
398C2×C4—241no—
399C2×C4D4×Dic311yescharacteristic
400C2×C4#93511yescharacteristic
401C2×C4#93911yescharacteristic
402C2×C4C2×#21711yescharacteristic
403C8—42no—
404C8—241no—
405C2×C3—21no—
406C2×C3—21no—
407C2×C3—41no—
408C2×C3—41no—
409C2×C3—82no—
410C2×C3#188211yescharacteristic
411C2×C3#189811yescharacteristic
412C2×C3#190611yescharacteristic
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—42no—
422C22—42no—
423C22—42no—
424C22—81no—
425C22—81no—
426C22—82no—
427C22—82no—
428C22#520211yescharacteristic
429C22#520511yescharacteristic
430C22#522011yescharacteristic
431C4—21no—
432C4—24no—
433C4—21no—
434C4—21no—
435C4—81no—
436C4—81no—
437C4—122no—
438C4—242no—
439C4—241no—
440C4—242no—
441C4C2×#83412yes—
442C3#936411yescharacteristic, complemented, Sylow
443C2—21no—
444C2—21no—
445C2—41no—
446C2—41no—
447C2—82no—
448C2#7167011yescharacteristic
449C2#7210011yescharacteristic
450C2#7211411yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#3007C2
socleC22×C3#386
prime radicalC22×C3×#41C22
cosocleC23×C3×C4C23
hypercenter#3007D3
hypocenterC3#9364
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3007

Complements

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

NH ≅ G/Ntype
4. #71701C2semidirect
5. #71688C2semidirect
6. #67684C2semidirect
7. #67683C2semidirect
10. C2×C3×#137C4semidirect
16. C3×#417C4semidirect
83. C2×#137Dic3semidirect
90. #417Dic3semidirect
442. C3#9364semidirect

Maximal subgroups

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

subgroupindex
C3×#30072
#840892
#721382
#717012
#716882
#676842
#676832
#93643