Finite Group Databasebeta

← #10694902

Subgroups — #10694902

Order 768.

510 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C2×#5048C211yesmaximal, characteristic
2C3×#1508C211yesmaximal, characteristic, complemented
3#71691C211yesmaximal, characteristic, complemented
4#69982C211yesmaximal, characteristic, complemented
5#69970C211yesmaximal, characteristic
6#69447C211yesmaximal, characteristic, complemented
7#69444C211yesmaximal, characteristic
8#9816—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C22×#215C411yescharacteristic, complemented
11C2×#930C2211yescharacteristic
12C23×C3⋊C8C2211yescharacteristic
13C3×#407C2211yescharacteristic
14C3×C4×C22⋊C4C2211yescharacteristic
15C2×#835C411yescharacteristic, complemented
16#5207C2211yescharacteristic
17C2×#830C2211yescharacteristic
18C2×#822—21no—
19#5048—21no—
20#5048—21no—
21#5048—21no—
22#5048—21no—
23C2×#809C411yescharacteristic, complemented
24C2×#808C411yescharacteristic, complemented
25C2×#803—21no—
26C3×#328—22no—
27#4558—21no—
28#4528—22no—
29#4524—21no—
30#4468—22no—
31#4451—22no—
32C2×#465—31no—
33#1810—31no—
34#1752—31no—
35#1508D311yescharacteristic, complemented
36#1507—31no—
37#1472—31no—
38#1471—31no—
39C22×C3×Q8C2×C411yescharacteristic
40C23×C3×C4C2311yescharacteristic
41C22×#67—21no—
42C22×D12C2×C411yescharacteristic
43C2×C3×M16—21no—
44C3×C22⋊Q8—21no—
45C3×C22⋊Q8—21no—
46C3×C22⋊Q8—21no—
47C3×C22⋊Q8—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4C2×C411yescharacteristic
50C2×C3×C22⋊C4—21no—
51C2×C3×C22⋊C4—21no—
52C2×C3×C42—21no—
53C2×#215—21no—
54C2×#215—21no—
55C2×#215—42no—
56#930—21no—
57#930—21no—
58#930—21no—
59#930—21no—
60C2×(C2×D3)⋊C4—21no—
61C2×C4⋊Dic3C2×C411yescharacteristic
62C2×#208—21no—
63C22×C3⋊C8—21no—
64C22×C3⋊C8—42no—
65C22×C3⋊C8C2×C411yescharacteristic
66C22×C3⋊C8C2×C411yescharacteristic
67C22×C3⋊C8D411yescharacteristic
68C22×C3⋊C8D411yescharacteristic
69C22×C3⋊C8D411yescharacteristic
70C22×C3⋊C8D411yescharacteristic
71C3×C22⋊C8—21no—
72C3×C22⋊C8—21no—
73C3×#95D412yes—
74#835—21no—
75#835—21no—
76#830—21no—
77#830—21no—
78#830—21no—
79#830D412yes—
80#822—42no—
81#809—21no—
82#809—21no—
83#808—21no—
84#808—21no—
85#803—42no—
86C22×SD16—31no—
87C23×C8—31no—
88C2×C22⋊Q8C2×D311yescharacteristic
89C2×#121—31no—
90#465—61no—
91#465—61no—
92#465—61no—
93#465—61no—
94C2×#108—61no—
95C2×#106—31no—
96C2×#103—31no—
97C2×#102—31no—
98#407—31no—
99#407C2×D311yescharacteristic
100C2×C22⋊C8—31no—
101C2×#97—61no—
102C4×C22⋊C4C2×D311yescharacteristic
103#341—61no—
104#336—61no—
105#328—22no—
106#327—62no—
107#324—62no—
108#323—62no—
109C24×C3#4111yescharacteristic
110C23×D3—21no—
111C2×C3×Q8—21no—
112C2×C3×Q8—21no—
113C2×C3×Q8—41no—
114C2×C3×Q8—41no—
115C22×C3×C4—21no—
116C22×C3×C4—22no—
117C22×C3×C4—21no—
118C22×C3×C4—41no—
119C22×C3×C4—41no—
120C22×C3×C4C22×C411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4#4111yescharacteristic
126C2×#67—41no—
127C2×#67—41no—
128C2×#67—82no—
129C22×Dic3—21no—
130C2×D12—21no—
131C2×D12—21no—
132C2×D12—41no—
133C2×D12—41no—
134C3×M16—41no—
135C3×M16—41no—
136C2×C3×C8—41no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—41no—
140C3×C4⋊C4—41no—
141C3×C22⋊C4—42no—
142C3×C22⋊C4—41no—
143C3×C22⋊C4—41no—
144C3×C42—41no—
145C3×C42—41no—
146#215—42no—
147#215—42no—
148(C2×D3)⋊C4—41no—
149(C2×D3)⋊C4—41no—
150C4⋊Dic3—21no—
151C4⋊Dic3—21no—
152#208—41no—
153#208—41no—
154C2×C3⋊C8—22no—
155C2×C3⋊C8—22no—
156C2×C3⋊C8—22no—
157C2×C3⋊C8—22no—
158C2×C3⋊C8—41no—
159C2×C3⋊C8—41no—
160C2×C3⋊C8—41no—
161C2×C3⋊C8—41no—
162C2×C3⋊C8—41no—
163C2×C3⋊C8—41no—
164C2×C3⋊C8C22⋊C411yescharacteristic
165C2×C3⋊C8C22⋊C411yescharacteristic
166C2×C3⋊C8C22⋊C411yescharacteristic
167C2×C3⋊C8C22⋊C411yescharacteristic
168C22×Q8C4×D311yescharacteristic
169C22×D4—31no—
170C22×D4—61no—
171C23×C4C22×D311yescharacteristic
172C2×SD16—61no—
173C2×SD16—61no—
174C2×SD16—122no—
175C2×M16—21no—
176C2×M16—61no—
177C22×C8—31no—
178C22×C8—31no—
179C22×C8—31no—
180C22×C8—31no—
181C22×C8—31no—
182C22×C8—31no—
183C22×C8—61no—
184C22×C8—122no—
185C22⋊Q8—21no—
186C22⋊Q8—21no—
187C22⋊Q8—21no—
188C22⋊Q8—21no—
189#121—61no—
190#121—61no—
191#121—61no—
192#121—61no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—31no—
195C2×C4⋊C4C4×D311yescharacteristic
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—61no—
199C2×C42—21no—
200#108—122no—
201#106—61no—
202#106—61no—
203#103—61no—
204#103—61no—
205#102—61no—
206#102—61no—
207C22⋊C8—21no—
208C22⋊C8—21no—
209C22⋊C8—32no—
210C22⋊C8—61no—
211C22⋊C8—61no—
212C22⋊C8—61no—
213#97—122no—
214#95#6712yes—
215C23×C3—22no—
216C23×C3—22no—
217C23×C3—41no—
218C23×C3C4×D411yescharacteristic
219C23×C3#12111yescharacteristic
220C23×C3#12411yescharacteristic
221C22×D3—41no—
222C22×D3—42no—
223C22×D3—81no—
224C22×D3—81no—
225C3×Q8—41no—
226C3×Q8—41no—
227C3×Q8—81no—
228C2×C3×C4—21no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4—42no—
238C2×C3×C4—42no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—42no—
242C2×C3×C4—42no—
243C2×C3×C4—81no—
244C2×C3×C4C2×C22⋊C411yescharacteristic
245C2×C3×C4C4×D411yescharacteristic
246C2×C3×C4#12111yescharacteristic
247C2×C3×C4#12711yescharacteristic
248#67—82no—
249#67—82no—
250C2×Dic3—41no—
251C2×Dic3—81no—
252D12—41no—
253D12—41no—
254D12—81no—
255C3×C8—81no—
256C3⋊C8—21no—
257C3⋊C8—21no—
258C3⋊C8—21no—
259C3⋊C8—42no—
260C3⋊C8—81no—
261C3⋊C8#9912yes—
262C24—61no—
263C24#23711yescharacteristic
264C2×Q8—21no—
265C2×Q8—21no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×D4—61no—
269C2×D4—61no—
270C2×D4—121no—
271C2×D4—121no—
272C2×D4—121no—
273C2×D4—121no—
274C2×D4—242no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—61no—
281C22×C4C2×C4×D311yescharacteristic
282C22×C4#23511yescharacteristic
283C22×C4D3×D411yescharacteristic
284C22×C4D3×D411yescharacteristic
285C22×C4D3×D411yescharacteristic
286C22×C4C2×#6711yescharacteristic
287SD16—122no—
288SD16—122no—
289M16—41no—
290M16—41no—
291M16—121no—
292M16—121no—
293C2×C8—31no—
294C2×C8—31no—
295C2×C8—31no—
296C2×C8—31no—
297C2×C8—41no—
298C2×C8—62no—
299C2×C8—62no—
300C2×C8—62no—
301C2×C8—62no—
302C2×C8—121no—
303C2×C8—121no—
304C2×C8—121no—
305C2×C8—121no—
306C2×C8—121no—
307C2×C8—121no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—61no—
313C4⋊C4—61no—
314C22⋊C4—42no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—121no—
318C22⋊C4—121no—
319C42—41no—
320C42—41no—
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—21no—
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—21no—
327C22×C3—21no—
328C22×C3—44no—
329C22×C3#39011yescharacteristic
330C22×C3#44412yes—
331C22×C3#47112yes—
332C2×D3—81no—
333C2×D3—81no—
334C2×D3—81no—
335C2×D3—162no—
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—82no—
340C3×C4—81no—
341C3×C4—81no—
342C3×C4C2×#9911yescharacteristic
343C3×C4#41011yescharacteristic
344Dic3—81no—
345C23—22no—
346C23—22no—
347C23—41no—
348C23—121no—
349C23—122no—
350C23—241no—
351C23—241no—
352C23#67⋊C411yescharacteristic
353C23#88511yescharacteristic
354C23#93911yescharacteristic
355Q8—41no—
356Q8—41no—
357Q8—81no—
358D4—121no—
359D4—121no—
360D4—241no—
361D4—242no—
362D4—242no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—81no—
379C2×C4—121no—
380C2×C4—241no—
381C2×C4D3×C22⋊C411yescharacteristic
382C2×C4#88411yescharacteristic
383C2×C4C4×#6711yescharacteristic
384C2×C4#94011yescharacteristic
385C8—32no—
386C8—61no—
387C8—61no—
388C8—61no—
389C8—81no—
390C8—122no—
391C8—241no—
392C2×C3—22no—
393C2×C3—22no—
394C2×C3—41no—
395C2×C3#196411yescharacteristic
396C2×C3#197511yescharacteristic
397C2×C3#197711yescharacteristic
398D3—161no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—44no—
407C22—241no—
408C22—241no—
409C22—241no—
410C22—482no—
411C22#482312yes—
412C22#495711yescharacteristic
413C22#520312yes—
414C4—21no—
415C4—21no—
416C4—21no—
417C4—82no—
418C4—81no—
419C4—81no—
420C4—241no—
421C4#474011yescharacteristic
422C4D3×#9911yescharacteristic
423C3#981611yescharacteristic, complemented, Sylow
424C2—22no—
425C2—22no—
426C2—41no—
427C2—481no—
428C2#7003011yescharacteristic
429C2#7033511yescharacteristic
430C2#7037411yescharacteristic
subgroupG/H
centerC22#4957
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1508C2
socleC22×C3#390
prime radicalC2×#930C22
cosocleC23×C3×C4C23
hypercenter#1508D3
hypocenterC3#9816
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1508

Complements

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

NH ≅ G/Ntype
2. C3×#1508C2semidirect
3. #71691C2semidirect
4. #69982C2semidirect
6. #69447C2semidirect
10. C22×#215C4semidirect
15. C2×#835C4semidirect
23. C2×#809C4semidirect
24. C2×#808C4semidirect
35. #1508D3semidirect
423. C3#9816semidirect

Maximal subgroups

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

subgroupindex
C2×#50482
C3×#15082
#716912
#699822
#699702
#694472
#694442
#98163