Finite Group Databasebeta

← #10692845

Subgroups — #10692845

Order 768.

568 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 91 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#79335C211yesmaximal, characteristic, complemented
2#73801C211yesmaximal, characteristic
3#70827C211yesmaximal, characteristic, complemented
4#70808C211yesmaximal, characteristic, complemented
5#70808C211yesmaximal, characteristic, complemented
6C3×#1285C211yesmaximal, characteristic, complemented
7#67621C211yesmaximal, characteristic, complemented
8#9447—31nomaximal, Sylow
9C2×#875C411yescharacteristic
10C2×#875C2211yescharacteristic
11C2×#874C411yescharacteristic
12C2×C4×D12C2211yescharacteristic
13C3×#432C2211yescharacteristic
14C3×#432C2211yescharacteristic
15C3×C4×C4⋊C4C2211yescharacteristic
16#4991—21no—
17#4991—21no—
18#4939C411yescharacteristic
19#4929C411yescharacteristic
20#4926C2211yescharacteristic
21#4926C2211yescharacteristic
22#4836—22no—
23#4726—21no—
24D3×C22⋊C8—21no—
25#4706—22no—
26#4665—21no—
27#4665—21no—
28#4468—21no—
29#4468—21no—
30#4464—22no—
31#4464—22no—
32#2973—31no—
33#2544—31no—
34#1533—31no—
35#1533—31no—
36#1466—31no—
37#1285—31no—
38#1285D311yescharacteristic, complemented
39C22×D12—21no—
40C22×D12—21no—
41C22×D12C2×C411yescharacteristic
42C22×D12C2×C411yescharacteristic
43C22×C4×D3D411yescharacteristic
44C22×C4×D3D411yescharacteristic
45C22×Dic6C2×C411yescharacteristic
46C2×C3×C4⋊C4—21no—
47C2×C3×C42C2×C411yescharacteristic
48C2×C3×C42C2×C411yescharacteristic
49C2×C3×C42C2311yescharacteristic
50C2×(C2×D3)⋊C4—21no—
51C2×(C2×D3)⋊C4—21no—
52C2×(C2×D3)⋊C4—21no—
53C2×(C2×D3)⋊C4—21no—
54C2×(C2×D3)⋊C4—21no—
55C2×(C2×D3)⋊C4—21no—
56C2×(C2×D3)⋊C4D411yescharacteristic
57C2×(C2×D3)⋊C4D411yescharacteristic
58C2×C4⋊Dic3C2×C411yescharacteristic
59C2×ΓL(1,𝔽25)—22no—
60C2×D3×C8—22no—
61#875—21no—
62#875—21no—
63#875—21no—
64#875—21no—
65#875—42no—
66#874—42no—
67C4×D12—21no—
68C4×D12—21no—
69C4×D12—21no—
70C4×D12—21no—
71C3×C4⋊C8—21no—
72C3×C4⋊C8—21no—
73C3×C22⋊C8—21no—
74C3×C22⋊C8—21no—
75C3×#97—21no—
76C3×#97D412yescomplemented
77C3×#95—21no—
78C3⋊#95—21no—
79C3⋊#95—21no—
80#830—21no—
81#830—21no—
82(C2×D3)⋊C8—22no—
83(C2×D3)⋊C8—22no—
84C4⋊(C3⋊C8)—21no—
85C4⋊(C3⋊C8)—21no—
86#803—21no—
87#803D412yescomplemented
88C2×#127—31no—
89C2×#124—31no—
90C2×#124—31no—
91C2×C4×D4—31no—
92#435—62no—
93#435—62no—
94#432—31no—
95#432—31no—
96#432C2×D311yescharacteristic
97#432C2×D311yescharacteristic
98#407—61no—
99C2×C22⋊C8—61no—
100#394—61no—
101#394—61no—
102#391—31no—
103#390—31no—
104#386—61no—
105#386—61no—
106C4×C4⋊C4C2×D311yescharacteristic
107#329—62no—
108#329—62no—
109#323—61no—
110#323—61no—
111C23×D3—21no—
112C23×D3C22⋊C411yescharacteristic
113C23×D3C22⋊C411yescharacteristic
114C22×C3×C4—21no—
115C22×C3×C4—21no—
116C22×C3×C4C22×C411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×Dic3—21no—
120C22×Dic3C22⋊C411yescharacteristic
121C22×Dic3C22⋊C411yescharacteristic
122C2×D12—22no—
123C2×D12—21no—
124C2×D12—21no—
125C2×D12—21no—
126C2×D12—21no—
127C2×D12—41no—
128C2×D12—41no—
129C2×D12—41no—
130C2×D12—42no—
131C2×D12—42no—
132C2×D12—42no—
133C2×D12—42no—
134C2×D12—81no—
135C2×D12—81no—
136C2×C4×D3—21no—
137C2×C4×D3—21no—
138C2×C4×D3—21no—
139C2×C4×D3—21no—
140C2×C4×D3—81no—
141C2×C4×D3—81no—
142C2×Dic6—21no—
143C2×Dic6—21no—
144C2×Dic6—41no—
145C2×Dic6—41no—
146C2×C3×C8—22no—
147C2×C3×C8—41no—
148C3×C4⋊C4—42no—
149C3×C42—41no—
150C3×C42—41no—
151C3×C42C2×D411yescharacteristic
152C3×C42C2×D411yescharacteristic
153C3×C42#4111yescharacteristic
154C3×C42#4111yescharacteristic
155(C2×D3)⋊C4—21no—
156(C2×D3)⋊C4—21no—
157(C2×D3)⋊C4—21no—
158(C2×D3)⋊C4—21no—
159(C2×D3)⋊C4—42no—
160(C2×D3)⋊C4—42no—
161(C2×D3)⋊C4—41no—
162(C2×D3)⋊C4—41no—
163(C2×D3)⋊C4—41no—
164(C2×D3)⋊C4—41no—
165(C2×D3)⋊C4—81no—
166(C2×D3)⋊C4—81no—
167C4⋊Dic3—21no—
168C4⋊Dic3—21no—
169C2×C3⋊C8—22no—
170C2×C3⋊C8—41no—
171ΓL(1,𝔽25)—82no—
172D3×C8—82no—
173C22×Q8—31no—
174C22×D4—31no—
175C22×D4—31no—
176C22×D4—61no—
177C22×D4—61no—
178C23×C4—31no—
179C23×C4—31no—
180C2×M16—62no—
181C22×C8—62no—
182#127—122no—
183#124—61no—
184#124—61no—
185#124—61no—
186#124—61no—
187#124—122no—
188C4×D4—61no—
189C4×D4—61no—
190C4×D4—61no—
191C4×D4—61no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—31no—
194C2×C22⋊C4—31no—
195C2×C22⋊C4—31no—
196C2×C22⋊C4—61no—
197C2×C22⋊C4—61no—
198C2×C22⋊C4—61no—
199C2×C22⋊C4—61no—
200C2×C22⋊C4—61no—
201C2×C22⋊C4—61no—
202C2×C42C4×D311yescharacteristic
203C2×C42C4×D311yescharacteristic
204C2×C42C22×D311yescharacteristic
205C4⋊C8—21no—
206C4⋊C8—21no—
207C4⋊C8—61no—
208C4⋊C8—61no—
209C22⋊C8—21no—
210C22⋊C8—21no—
211C22⋊C8—62no—
212C22⋊C8—62no—
213C22⋊C8—61no—
214C22⋊C8—61no—
215#97—21no—
216#97—32no—
217#97—61no—
218#97D1212yescomplemented
219#95—21no—
220#95—61no—
221#95—61no—
222C23×C3C2×C22⋊C411yescharacteristic
223C22×D3—21no—
224C22×D3—21no—
225C22×D3—21no—
226C22×D3—21no—
227C22×D3—21no—
228C22×D3—41no—
229C22×D3—41no—
230C22×D3—41no—
231C22×D3—41no—
232C22×D3—41no—
233C22×D3—41no—
234C22×D3—41no—
235C22×D3—41no—
236C22×D3—41no—
237C22×D3—41no—
238C22×D3—41no—
239C22×D3#10012yes—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—22no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4C4×D411yescharacteristic
251C2×C3×C4C4×D411yescharacteristic
252C2×C3×C4#12011yescharacteristic
253C2×C3×C4#12111yescharacteristic
254C2×C3×C4#12111yescharacteristic
255C2×C3×C4#12311yescharacteristic
256C2×Dic3—21no—
257C2×Dic3—21no—
258C2×Dic3—41no—
259C2×Dic3—41no—
260C2×Dic3—41no—
261C2×Dic3—41no—
262C2×Dic3—41no—
263D12—42no—
264D12—42no—
265D12—81no—
266D12—81no—
267D12—84no—
268D12—84no—
269C4×D3—42no—
270C4×D3—42no—
271C4×D3—81no—
272C4×D3—81no—
273Dic6—81no—
274Dic6—81no—
275C3×C8—42no—
276C3×C8—81no—
277C3⋊C8—42no—
278C3⋊C8—81no—
279C24—31no—
280C24—31no—
281C24—61no—
282C2×Q8—61no—
283C2×Q8—61no—
284C2×Q8—121no—
285C2×Q8—121no—
286C2×D4—62no—
287C2×D4—61no—
288C2×D4—61no—
289C2×D4—61no—
290C2×D4—61no—
291C2×D4—121no—
292C2×D4—121no—
293C2×D4—121no—
294C2×D4—122no—
295C2×D4—122no—
296C2×D4—122no—
297C2×D4—122no—
298C2×D4—241no—
299C2×D4—241no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—31no—
303C22×C4—31no—
304C22×C4—61no—
305C22×C4—61no—
306C22×C4—61no—
307C22×C4—61no—
308C22×C4—61no—
309C22×C4—241no—
310C22×C4—241no—
311C22×C4C2×C4×D311yescharacteristic
312C22×C4D3×D411yescharacteristic
313C22×C4D3×D411yescharacteristic
314M16—242no—
315C2×C8—22no—
316C2×C8—41no—
317C2×C8—62no—
318C2×C8—121no—
319C2×C8—242no—
320C4⋊C4—42no—
321C4⋊C4—61no—
322C4⋊C4—61no—
323C22⋊C4—61no—
324C22⋊C4—61no—
325C22⋊C4—61no—
326C22⋊C4—61no—
327C22⋊C4—122no—
328C22⋊C4—122no—
329C22⋊C4—121no—
330C22⋊C4—121no—
331C22⋊C4—121no—
332C22⋊C4—121no—
333C22⋊C4—241no—
334C22⋊C4—241no—
335C42—41no—
336C42—41no—
337C42C2×D1211yescharacteristic
338C42#23511yescharacteristic
339C42D3×D411yescharacteristic
340C42#23911yescharacteristic
341C22×C3—21no—
342C22×C3—21no—
343C22×C3#38611yescharacteristic
344C22×C3C2×#10011yescharacteristic
345C22×C3#41311yescharacteristic
346C2×D3—22no—
347C2×D3—41no—
348C2×D3—41no—
349C2×D3—42no—
350C2×D3—42no—
351C2×D3—41no—
352C2×D3—41no—
353C2×D3—41no—
354C2×D3—81no—
355C2×D3—81no—
356C2×D3—81no—
357C2×D3—81no—
358C2×D3—81no—
359C2×D3—81no—
360C2×D3—81no—
361C2×D3—81no—
362C2×D3—81no—
363C3×C4—21no—
364C3×C4—21no—
365C3×C4—21no—
366C3×C4—21no—
367C3×C4—21no—
368C3×C4—21no—
369C3×C4—42no—
370C3×C4—81no—
371Dic3—41no—
372Dic3—41no—
373Dic3—81no—
374C23—32no—
375C23—61no—
376C23—61no—
377C23—61no—
378C23—61no—
379C23—61no—
380C23—121no—
381C23—121no—
382C23—121no—
383C23—121no—
384C23—121no—
385C23—121no—
386C23—121no—
387C23—121no—
388C23—121no—
389C23—121no—
390C23—121no—
391C23D3×C22⋊C411yescharacteristic
392Q8—241no—
393Q8—241no—
394D4—122no—
395D4—122no—
396D4—241no—
397D4—241no—
398D4—244no—
399D4—244no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—22no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—61no—
411C2×C4—61no—
412C2×C4—122no—
413C2×C4—122no—
414C2×C4—121no—
415C2×C4—121no—
416C2×C4—121no—
417C2×C4—121no—
418C2×C4—121no—
419C2×C4—241no—
420C2×C4—241no—
421C2×C4C4×D1211yescharacteristic
422C2×C4#88211yescharacteristic
423C2×C4#88411yescharacteristic
424C2×C4#89311yescharacteristic
425C2×C4#89411yescharacteristic
426C2×C4#89511yescharacteristic
427C8—42no—
428C8—81no—
429C8—122no—
430C8—241no—
431C2×C3—21no—
432C2×C3—21no—
433C2×C3#188411yescharacteristic
434C2×C3#190011yescharacteristic
435C2×C3#191111yescharacteristic
436D3—42no—
437D3—81no—
438D3—161no—
439C22—21no—
440C22—21no—
441C22—62no—
442C22—122no—
443C22—122no—
444C22—121no—
445C22—121no—
446C22—121no—
447C22—121no—
448C22—121no—
449C22—241no—
450C22—241no—
451C22—241no—
452C22—241no—
453C22—241no—
454C22—241no—
455C22—241no—
456C22—241no—
457C22—241no—
458C22#466511yescharacteristic
459C22D3×#10011yescharacteristic
460C22#475111yescharacteristic
461C4—21no—
462C4—21no—
463C4—21no—
464C4—21no—
465C4—21no—
466C4—21no—
467C4—42no—
468C4—81no—
469C4—121no—
470C4—121no—
471C4—241no—
472C3#944711yescharacteristic, complemented, Sylow
473C2—21no—
474C2—21no—
475C2—122no—
476C2—241no—
477C2—481no—
478C2#6870511yescharacteristic
479C2#6877711yescharacteristic
480C2#6899211yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1285C2
socleC22×C3#386
prime radicalC2×#874C4
cosocleC2×C3×C42C23
hypercenter#1285D3
hypocenterC3#9447
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1285

Complements

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

NH ≅ G/Ntype
1. #79335C2semidirect
3. #70827C2semidirect
4. #70808C2semidirect
5. #70808C2semidirect
6. C3×#1285C2semidirect
7. #67621C2semidirect
38. #1285D3semidirect
76. C3×#97D4semidirect
87. #803D4semidirect
218. #97D12semidirect
472. C3#9447semidirect

Maximal subgroups

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

subgroupindex
#793352
#738012
#708272
#708082
C3×#12852
#676212
#94473