Finite Group Databasebeta

← #10692369

Subgroups — #10692369

Order 768.

732 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 111 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×#2341C211yesmaximal, characteristic
2#84084C211yesmaximal, characteristic
3#72138C211yesmaximal, characteristic
4#69529C211yesmaximal, characteristic, complemented
5#69445C211yesmaximal, characteristic, complemented
6#67626C211yesmaximal, characteristic, complemented
7#67618C211yesmaximal, characteristic, complemented
8#9364—31nomaximal, Sylow
9C22×C3×#41C2211yescharacteristic
10C2×C3×C4×Q8—21no—
11C2×#951C412yescomplemented
12C2×#924C2211yescharacteristic
13C3×#390—21no—
14C3×#390—22no—
15C3×#380—21no—
16C3×#380C2211yescharacteristic
17C3×C4×C22⋊C4—21no—
18C3×C4×C22⋊C4C2211yescharacteristic
19#5239—22no—
20C2×#837—22no—
21#5232C412yescomplemented
22#5228—21no—
23#5218—21no—
24C4×#217C2211yescharacteristic
25#5207C2211yescharacteristic
26#5207C2211yescharacteristic
27#4527—22no—
28#4452—21no—
29#4452—21no—
30#3007—31no—
31#2341D311yescharacteristic
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×Q8D411yescharacteristic
43C22×C3×D4—21no—
44C22×C3×D4D411yescharacteristic
45C23×C3×C4—21no—
46C23×C3×C4C2311yescharacteristic
47C22×Dic6C2×C411yescharacteristic
48C3×C4×Q8—44no—
49C2×C3×C4⋊C4—22no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C22⋊C4—22no—
52C2×C3×C22⋊C4—21no—
53C2×C3×C22⋊C4—22no—
54C2×C3×C22⋊C4D412yes—
55C2×C3×C42—22no—
56C2×C3×C42—21no—
57C2×#217—21no—
58C2×#217—21no—
59#951—44no—
60C2×#216—22no—
61C2×#214—22no—
62C2×C4⋊Dic3C2×C411yescharacteristic
63#924—21no—
64#924—21no—
65#924—21no—
66#924—21no—
67C2×Dic3⋊C4—21no—
68C2×C4×Dic3—21no—
69C2×#208—21no—
70C2×#208C2×C412yes—
71C3×#95—21no—
72C3×#95D411yescharacteristic
73C3×#95D411yescharacteristic
74#837—44no—
75#835—22no—
76#832—22no—
77C3⋊#95—21no—
78#830—21no—
79#830—21no—
80#830—21no—
81#830—21no—
82C22×#41C2×D311yescharacteristic
83C2×#137—32no—
84C2×C22⋊Q8—31no—
85C2×C4×Q8—21no—
86#452—62no—
87#451—61no—
88#450—61no—
89C2×#104—62no—
90#417—32no—
91#407—31no—
92#407—31no—
93#390—21no—
94#390—22no—
95#380—21no—
96#380C2×D311yescharacteristic
97C4×C22⋊C4—21no—
98C4×C22⋊C4—31no—
99C4×C22⋊C4C2×D311yescharacteristic
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×Q8C22⋊C412yes—
114C2×C3×D4—22no—
115C2×C3×D4—22no—
116C2×C3×D4—21no—
117C2×C3×D4—42no—
118C2×C3×D4—44no—
119C2×C3×D4C22⋊C412yes—
120C22×C3×C4—21no—
121C22×C3×C4—21no—
122C22×C3×C4—22no—
123C22×C3×C4—22no—
124C22×C3×C4—22no—
125C22×C3×C4—21no—
126C22×C3×C4—42no—
127C22×C3×C4—44no—
128C22×C3×C4C22×C411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4#4111yescharacteristic
133C22×C3×C4#4111yescharacteristic
134C22×Dic3—21no—
135C22×Dic3—21no—
136C22×Dic3—41no—
137C2×Dic6—21no—
138C2×Dic6—21no—
139C2×Dic6—42no—
140C3×C4⋊C4—44no—
141C3×C4⋊C4—42no—
142C3×C22⋊C4—42no—
143C3×C22⋊C4—44no—
144C3×C22⋊C4—42no—
145C3×C22⋊C4—82no—
146C3×C42—44no—
147C3×C42—42no—
148#217—22no—
149#217—41no—
150#217—41no—
151#217—41no—
152#216—44no—
153#214—44no—
154C4⋊Dic3—21no—
155C4⋊Dic3—21no—
156Dic3⋊C4—41no—
157Dic3⋊C4—41no—
158C4×Dic3—42no—
159#208—42no—
160#208—42no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—41no—
163C2×#41—21no—
164C2×#41—21no—
165C2×#41—44no—
166C2×#41C4×D312yes—
167C2×#41D1212yes—
168C22×Q8—31no—
169C22×Q8#6711yescharacteristic
170C22×D4—21no—
171C22×D4#6711yescharacteristic
172C23×C4—21no—
173C23×C4C22×D311yescharacteristic
174#137—124no—
175C2×Q16—62no—
176C2×SD16—62no—
177C2×M16—32no—
178C2×M16—61no—
179C22⋊Q8—61no—
180C22⋊Q8—61no—
181C22⋊Q8—61no—
182C22⋊Q8—61no—
183C4×Q8—44no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—31no—
187C2×C4⋊C4—61no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—61no—
192C2×C22⋊C4—61no—
193C2×C22⋊C4#6712yes—
194C2×C42—22no—
195C2×C42—21no—
196C2×C42—61no—
197#104—124no—
198#103—62no—
199#102—62no—
200C22⋊C8—61no—
201C22⋊C8—61no—
202C22⋊C8—61no—
203C22⋊C8—61no—
204#95—21no—
205#95—61no—
206#95#6711yescharacteristic
207#95#6711yescharacteristic
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#9914yes—
223C3×D4—22no—
224C3×D4—22no—
225C3×D4—44no—
226C3×D4—44no—
227C3×D4—42no—
228C2×C3×C4—24no—
229C2×C3×C4—22no—
230C2×C3×C4—22no—
231C2×C3×C4—22no—
232C2×C3×C4—22no—
233C2×C3×C4—22no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4—44no—
238C2×C3×C4—44no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—42no—
242C2×C3×C4—41no—
243C2×C3×C4—42no—
244C2×C3×C4—41no—
245C2×C3×C4—42no—
246C2×C3×C4—41no—
247C2×C3×C4C2×C22⋊C411yescharacteristic
248C2×C3×C4C4×D411yescharacteristic
249C2×C3×C4#12111yescharacteristic
250C2×C3×C4#12311yescharacteristic
251C2×Dic3—42no—
252C2×Dic3—41no—
253C2×Dic3—41no—
254C2×Dic3—81no—
255C2×Dic3—81no—
256C2×Dic3—81no—
257C2×Dic3—81no—
258Dic6—42no—
259Dic6—81no—
260C3⋊C8—42no—
261C3⋊C8—81no—
262C24—21no—
263C24C2×#6711yescharacteristic
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—61no—
273C2×Q8—61no—
274C2×Q8—122no—
275C2×Q8(C2×D3)⋊C412yes—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—21no—
279C2×D4—42no—
280C2×D4—44no—
281C2×D4(C2×D3)⋊C412yes—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—42no—
289C22×C4—44no—
290C22×C4—61no—
291C22×C4—61no—
292C22×C4—121no—
293C22×C4C2×C4×D311yescharacteristic
294C22×C4C2×D1211yescharacteristic
295C22×C4#23511yescharacteristic
296C22×C4#23511yescharacteristic
297C22×C4C2×#6711yescharacteristic
298C22×C4C2×#6711yescharacteristic
299Q16—124no—
300SD16—124no—
301M16—122no—
302M16—122no—
303C2×C8—62no—
304C2×C8—121no—
305C4⋊C4—44no—
306C4⋊C4—42no—
307C4⋊C4—61no—
308C4⋊C4—61no—
309C4⋊C4—121no—
310C4⋊C4—121no—
311C22⋊C4—42no—
312C22⋊C4—44no—
313C22⋊C4—42no—
314C22⋊C4—62no—
315C22⋊C4—82no—
316C22⋊C4—121no—
317C22⋊C4—121no—
318C22⋊C4—121no—
319C42—44no—
320C42—42no—
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—21no—
342C3×C4—24no—
343C3×C4—21no—
344C3×C4—42no—
345C3×C4—82no—
346C3×C4—81no—
347C3×C4C2×#9912yes—
348Dic3—82no—
349Dic3—81no—
350Dic3—81no—
351C23—21no—
352C23—21no—
353C23—22no—
354C23—41no—
355C23—41no—
356C23—42no—
357C23—42no—
358C23—41no—
359C23—41no—
360C23C4×#6711yescharacteristic
361C23#93011yescharacteristic
362C23#95311yescharacteristic
363Q8—24no—
364Q8—22no—
365Q8—122no—
366Q8—241no—
367Q8#80614yes—
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—22no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—122no—
393C2×C4—121no—
394C2×C4—121no—
395C2×C4—241no—
396C2×C4—241no—
397C2×C4—241no—
398C2×C4—241no—
399C2×C4C2×(C2×D3)⋊C411yescharacteristic
400C2×C4C4×#6711yescharacteristic
401C2×C4#92911yescharacteristic
402C2×C4#93011yescharacteristic
403C8—122no—
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#521311yescharacteristic
429C22#524211yescharacteristic
430C22#524611yescharacteristic
431C4—21no—
432C4—24no—
433C4—21no—
434C4—21no—
435C4—42no—
436C4—82no—
437C4—81no—
438C4—242no—
439C4—241no—
440C4—241no—
441C4C2×#80612yes—
442C3#936411yescharacteristic, complemented, Sylow
443C2—21no—
444C2—21no—
445C2—41no—
446C2—41no—
447C2—82no—
448C2#7154311yescharacteristic
449C2#7164511yescharacteristic
450C2#7166611yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2341C2
socleC22×C3#386
prime radicalC22×C3×#41C22
cosocleC23×C3×C4C23
hypercenter#2341D3
hypocenterC3#9364
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2341

Complements

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

NH ≅ G/Ntype
4. #69529C2semidirect
5. #69445C2semidirect
6. #67626C2semidirect
7. #67618C2semidirect
11. C2×#951C4semidirect
21. #5232C4semidirect
442. C3#9364semidirect

Maximal subgroups

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

subgroupindex
C3×#23412
#840842
#721382
#695292
#694452
#676262
#676182
#93643