Finite Group Databasebeta

← #10686707

Subgroups — #10686707

Order 768.

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

normal subgroups only

This group has 726 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
1#85745C211yesmaximal, characteristic
2C3×#1750C211yesmaximal, characteristic, complemented
3#71572C211yesmaximal, characteristic, complemented
4#69799C212yesmaximal, complemented
5#69736C212yesmaximal, complemented
6#8348—31nomaximal, Sylow
7C2×C3×#117C2211yescharacteristic
8C2×#929—21no—
9#5531—22no—
10#5531—22no—
11#5527—22no—
12#5527—22no—
13#5515—21no—
14#5515C412yes—
15#5493C411yescharacteristic
16#5486—21no—
17#5483C411yescharacteristic
18#5479—21no—
19#5479C2212yes—
20C2×C3×C22⋊C8—21no—
21C2×C3×#97C2211yescharacteristic, complemented
22C2×C3×C4×C8C2211yescharacteristic, complemented
23#4926C2212yescomplemented
24C3×#336—21no—
25#4528—21no—
26#4525—21no—
27#4465—22no—
28#4464—22no—
29#4459—22no—
30#4456—22no—
31#3491—31no—
32#1750D311yescharacteristic, complemented
33#1532—32no—
34#1494—32no—
35#1464—31no—
36C23×C3×C4D411yescharacteristic
37C22×#67—21no—
38C22×C3×C8—21no—
39C22×C3×C8—21no—
40C3×#117—22no—
41C3×#117—22no—
42C2×C3×C4⋊C4C2×C411yescharacteristic
43C2×C3×C4⋊C4D411yescharacteristic
44C2×C3×C22⋊C4C2×C411yescharacteristic
45C2×C3×C22⋊C4D411yescharacteristic
46C2×C3×C42D411yescharacteristic
47C2×C3×C42C2311yescharacteristic
48C2×#217—21no—
49#930—21no—
50#930C2×C412yes—
51#929—22no—
52#929—22no—
53#929—22no—
54#929—22no—
55C4×#67—22no—
56C4×#67—22no—
57C2×(C2×D3)⋊C4—21no—
58C2×(C2×D3)⋊C4—21no—
59#924—21no—
60#924C2×C412yes—
61C2×Dic3⋊C4—21no—
62C2×Dic3⋊C4—21no—
63#898—22no—
64#898—22no—
65#896—22no—
66#896—22no—
67#894—22no—
68#894—22no—
69#889—22no—
70#889—22no—
71#886—22no—
72#882—22no—
73#67⋊C4—22no—
74#67⋊C4—22no—
75D3×C22⋊C4—22no—
76#878—22no—
77#878—22no—
78#877—22no—
79#876—22no—
80#876—22no—
81#875—21no—
82#875D412yes—
83C4×D12—21no—
84C4×D12D412yes—
85#872—22no—
86#872—22no—
87#870—21no—
88#870D412yes—
89C4×Dic6—21no—
90C4×Dic6D412yes—
91C3×C22⋊C8—24no—
92C3×#97—22no—
93C3×C4×C8—22no—
94#830—22no—
95C4⋊(C3⋊C8)—22no—
96#803—22no—
97#552—62no—
98#552—62no—
99#551—62no—
100#551—62no—
101#543—31no—
102#539—32no—
103#539—61no—
104#534—31no—
105#529—32no—
106#529—61no—
107C2×#123—61no—
108#518—61no—
109C2×#117C2×D311yescharacteristic
110#432—32no—
111C2×C22⋊C8—21no—
112C2×#97C2×D311yescharacteristic, complemented
113C2×C4×C8C2×D311yescharacteristic, complemented
114#336—21no—
115#330—62no—
116#329—62no—
117Q8⋊C8—62no—
118D4⋊C8—62no—
119#324—61no—
120#323—61no—
121C24×C3C22⋊C411yescharacteristic
122C23×D3—21no—
123C22×C3×C4—22no—
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4—41no—
127C22×C3×C4C22⋊C411yescharacteristic
128C22×C3×C4C22⋊C411yescharacteristic
129C22×C3×C4C22⋊C411yescharacteristic
130C22×C3×C4C22×C411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C2×#67—22no—
134C2×#67—22no—
135C2×#67—42no—
136C2×#67—42no—
137C22×Dic3—21no—
138C22×Dic3—21no—
139C22×Dic3—21no—
140C2×D12—21no—
141C2×D12C22⋊C412yes—
142C2×C4×D3—22no—
143C2×C4×D3—22no—
144C2×Dic6—21no—
145C2×Dic6C22⋊C412yes—
146C2×C3×C8—22no—
147C2×C3×C8—22no—
148C2×C3×C8—41no—
149C2×C3×C8—41no—
150C2×C3×C8—41no—
151C2×C3×C8—41no—
152C3×C4⋊C4—22no—
153C3×C4⋊C4—22no—
154C3×C22⋊C4—22no—
155C3×C22⋊C4—22no—
156C3×C42—21no—
157C3×C42—21no—
158C3×C42C2×D412yes—
159C3×C42C2×D412yes—
160#217—22no—
161#217—22no—
162(C2×D3)⋊C4—22no—
163(C2×D3)⋊C4—22no—
164(C2×D3)⋊C4—22no—
165(C2×D3)⋊C4—22no—
166(C2×D3)⋊C4—22no—
167(C2×D3)⋊C4—22no—
168(C2×D3)⋊C4—22no—
169(C2×D3)⋊C4—22no—
170C4⋊Dic3—21no—
171C4⋊Dic3—21no—
172C4⋊Dic3C22⋊C412yes—
173C4⋊Dic3C22⋊C412yes—
174Dic3⋊C4—22no—
175Dic3⋊C4—22no—
176Dic3⋊C4—22no—
177Dic3⋊C4—22no—
178Dic3⋊C4—22no—
179Dic3⋊C4—22no—
180Dic3⋊C4—22no—
181Dic3⋊C4—22no—
182C4×Dic3—22no—
183C4×Dic3—22no—
184C2×C3⋊C8—42no—
185C22×D4—61no—
186C23×C4#6711yescharacteristic
187C22×C8—21no—
188C22×C8—21no—
189#126—62no—
190#126—62no—
191#126—62no—
192#126—62no—
193#125—32no—
194#125—61no—
195#125—62no—
196#125—62no—
197#124—32no—
198#124—61no—
199#123—62no—
200#123—62no—
201#123—62no—
202#123—62no—
203#123—62no—
204#123—62no—
205#123—62no—
206C22⋊Q8—32no—
207C22⋊Q8—62no—
208C22⋊Q8—62no—
209C22⋊Q8—62no—
210C22⋊Q8—62no—
211C22⋊Q8—61no—
212#121—32no—
213#121—61no—
214#120—62no—
215C4×Q8—32no—
216C4×Q8—61no—
217C4×D4—32no—
218C4×D4—61no—
219C4×D4—62no—
220C4×D4—62no—
221C4×D4—62no—
222C4×D4—62no—
223#117—22no—
224#117—22no—
225#117—62no—
226#117—62no—
227#117—62no—
228C2×C4⋊C4—61no—
229C2×C4⋊C4—61no—
230C2×C4⋊C4C4×D311yescharacteristic
231C2×C4⋊C4D1211yescharacteristic
232C2×C22⋊C4—62no—
233C2×C22⋊C4—61no—
234C2×C22⋊C4—61no—
235C2×C22⋊C4—61no—
236C2×C22⋊C4C4×D311yescharacteristic
237C2×C22⋊C4D1211yescharacteristic
238C2×C42#6711yescharacteristic
239C2×C42C22×D311yescharacteristic
240C4⋊C8—62no—
241C22⋊C8—24no—
242C22⋊C8—62no—
243#97—22no—
244#97—62no—
245C4×C8—22no—
246C23×C3—21no—
247C23×C3—21no—
248C23×C3—41no—
249C23×C3—41no—
250C23×C3#10412yes—
251C23×C3C2×C22⋊C411yescharacteristic
252C22×D3—22no—
253C22×D3—42no—
254C22×D3—81no—
255C22×D3—81no—
256C2×C3×C4—21no—
257C2×C3×C4—21no—
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—41no—
264C2×C3×C4—41no—
265C2×C3×C4—42no—
266C2×C3×C4—42no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4C2×C22⋊C412yes—
270C2×C3×C4#12012yes—
271C2×C3×C4#12011yescharacteristic
272C2×C3×C4#12011yescharacteristic
273#67—44no—
274#67—42no—
275#67—42no—
276C2×Dic3—22no—
277C2×Dic3—22no—
278C2×Dic3—22no—
279C2×Dic3—42no—
280C2×Dic3—81no—
281C2×Dic3—81no—
282D12—42no—
283D12—81no—
284C4×D3—82no—
285C4×D3—82no—
286Dic6—42no—
287Dic6—81no—
288C3×C8—42no—
289C3×C8—42no—
290C3⋊C8—82no—
291C24—61no—
292C24(C2×D3)⋊C411yescharacteristic
293C2×Q8—32no—
294C2×Q8—61no—
295C2×D4—32no—
296C2×D4—61no—
297C2×D4—62no—
298C2×D4—62no—
299C2×D4—122no—
300C2×D4—122no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—41no—
305C22×C4—62no—
306C22×C4—61no—
307C22×C4—61no—
308C22×C4—61no—
309C22×C4—62no—
310C22×C4(C2×D3)⋊C411yescharacteristic
311C22×C4(C2×D3)⋊C411yescharacteristic
312C22×C4(C2×D3)⋊C411yescharacteristic
313C22×C4C2×C4×D311yescharacteristic
314C22×C4C2×D1211yescharacteristic
315C22×C4C2×#6711yescharacteristic
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—122no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—32no—
326C4⋊C4—32no—
327C4⋊C4—62no—
328C4⋊C4—62no—
329C4⋊C4—62no—
330C4⋊C4—62no—
331C4⋊C4—61no—
332C4⋊C4—61no—
333C4⋊C4—62no—
334C4⋊C4—62no—
335C4⋊C4—62no—
336C4⋊C4—62no—
337C22⋊C4—22no—
338C22⋊C4—22no—
339C22⋊C4—62no—
340C22⋊C4—62no—
341C22⋊C4—62no—
342C22⋊C4—62no—
343C22⋊C4—62no—
344C22⋊C4—62no—
345C22⋊C4—62no—
346C22⋊C4—62no—
347C22⋊C4—62no—
348C22⋊C4—62no—
349C42—21no—
350C42—21no—
351C42—62no—
352C42—62no—
353C42D3×D412yes—
354C42D3×D412yes—
355C22×C3—21no—
356C22×C3—21no—
357C22×C3—22no—
358C22×C3—42no—
359C22×C3—41no—
360C22×C3—41no—
361C22×C3—81no—
362C22×C3#37911yescharacteristic
363C22×C3C2×#10411yescharacteristic
364C22×C3#42111yescharacteristic
365C2×D3—42no—
366C2×D3—42no—
367C2×D3—42no—
368C2×D3—162no—
369C3×C4—22no—
370C3×C4—21no—
371C3×C4—21no—
372C3×C4—22no—
373C3×C4—41no—
374C3×C4—41no—
375C3×C4—41no—
376Dic3—42no—
377Dic3—42no—
378Dic3—42no—
379C23—21no—
380C23—21no—
381C23—41no—
382C23—41no—
383C23—62no—
384C23—122no—
385C23—241no—
386C23—241no—
387C23#80512yes—
388C23C2×(C2×D3)⋊C411yescharacteristic
389Q8—122no—
390Q8—241no—
391D4—122no—
392D4—124no—
393D4—122no—
394D4—122no—
395D4—241no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—62no—
410C2×C4—62no—
411C2×C4—62no—
412C2×C4—122no—
413C2×C4—242no—
414C2×C4—242no—
415C2×C4—241no—
416C2×C4—241no—
417C2×C4D3×C22⋊C412yes—
418C2×C4#88212yes—
419C2×C4#93711yescharacteristic
420C2×C4#93711yescharacteristic
421C8—42no—
422C8—42no—
423C8—242no—
424C2×C3—21no—
425C2×C3—21no—
426C2×C3—42no—
427C2×C3#178911yescharacteristic
428C2×C3#179111yescharacteristic
429C2×C3#179411yescharacteristic
430D3—82no—
431C22—21no—
432C22—21no—
433C22—22no—
434C22—42no—
435C22—41no—
436C22—41no—
437C22—81no—
438C22—122no—
439C22—122no—
440C22—122no—
441C22—482no—
442C22C2×#80511yescharacteristic
443C22#495511yescharacteristic
444C22#500511yescharacteristic
445C4—21no—
446C4—21no—
447C4—22no—
448C4—22no—
449C4—41no—
450C4—41no—
451C4—41no—
452C4—122no—
453C4—122no—
454C4—122no—
455C3#834811yescharacteristic, complemented, Sylow
456C2—21no—
457C2—21no—
458C2—42no—
459C2—242no—
460C2#6945411yescharacteristic
461C2#7000911yescharacteristic
462C2#7001011yescharacteristic
subgroupG/H
centerC22#4955
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1750C2
socleC22×C3#379
prime radical#5493C4
cosocleC2×C3×C42C23
hypercenter#1750D3
hypocenterC3#8348
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1750

Complements

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

NH ≅ G/Ntype
2. C3×#1750C2semidirect
3. #71572C2semidirect
4. #69799C2semidirect
5. #69736C2semidirect
21. C2×C3×#97C22semidirect
22. C2×C3×C4×C8C22semidirect
23. #4926C22semidirect
32. #1750D3semidirect
112. C2×#97C2×D3semidirect
113. C2×C4×C8C2×D3semidirect
455. C3#8348semidirect

Maximal subgroups

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

subgroupindex
#857452
C3×#17502
#715722
#697992
#697362
#83483