Finite Group Databasebeta

← #10695519

Subgroups — #10695519

Order 768.

568 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 85 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#78642C211yesmaximal, characteristic
2#78624C211yesmaximal, characteristic
3#73951C211yesmaximal, characteristic, complemented
4#69980C211yesmaximal, characteristic, complemented
5#69975C211yesmaximal, characteristic, complemented
6#69528C211yesmaximal, characteristic
7C3×#1305C211yesmaximal, characteristic, complemented
8#9933—31nomaximal, Sylow
9C2×#930C2211yescharacteristic
10C2×#926C2211yescharacteristic
11C2×#924C2211yescharacteristic
12C3×#407C2211yescharacteristic
13C3×#407C2211yescharacteristic
14C3×#380C2211yescharacteristic
15C4×#217C2211yescharacteristic
16#5134—22no—
17#5131—21no—
18#5121—21no—
19#4984—21no—
20#4973—21no—
21C2×#806—22no—
22#4953—22no—
23#4738—21no—
24#4737—22no—
25#4736—21no—
26#4736—21no—
27#4735—21no—
28#4731—21no—
29#4722—21no—
30#4661—21no—
31#4659—21no—
32#4555—22no—
33#4467—22no—
34#3193—31no—
35#3094—31no—
36#2625—31no—
37#1964—31no—
38#1826—31no—
39#1508—31no—
40#1305D311yescharacteristic, complemented
41C23×C3×C4C2311yescharacteristic
42C22×#67—21no—
43C22×D12D411yescharacteristic
44C22×Dic6D411yescharacteristic
45C2×C3×M16—21no—
46C2×C3×M16D412yes—
47C2×C3×C4⋊C4—21no—
48C2×C3×C22⋊C4D412yes—
49C2×#217—21no—
50C2×#217—21no—
51C2×#217—21no—
52#930—21no—
53#930—21no—
54#930—21no—
55#930—21no—
56C2×(C2×D3)⋊C4—21no—
57#926—21no—
58#926—21no—
59#926—21no—
60#926—21no—
61C2×C4⋊Dic3—22no—
62C2×C4⋊Dic3D411yescharacteristic
63C2×C4⋊Dic3D411yescharacteristic
64C2×C4⋊Dic3D411yescharacteristic
65C2×C4⋊Dic3D411yescharacteristic
66#924—21no—
67#924—21no—
68#924—21no—
69#924—21no—
70C2×Dic3⋊C4—21no—
71C2×C4×Dic3—21no—
72C2×C4×Dic3—21no—
73C2×#204—22no—
74C3×C22⋊C8—21no—
75C3×C22⋊C8—21no—
76C3×C22⋊C8—21no—
77C3×C22⋊C8D412yes—
78C3×#95—21no—
79C3⋊#95—22no—
80C3⋊#95—22no—
81C3⋊#95—22no—
82C3⋊#95—21no—
83#821—22no—
84#821—22no—
85#821—22no—
86#818—22no—
87#818—22no—
88#817—22no—
89#816—22no—
90#806—44no—
91C2×C22⋊Q8—31no—
92C2×#121—31no—
93C2×#117—31no—
94#484—61no—
95#482—62no—
96#481—61no—
97#480—61no—
98#480—61no—
99#468—62no—
100#450—61no—
101#448—61no—
102#428—61no—
103#418—61no—
104C2×#99—62no—
105#407C2×D311yescharacteristic
106#407C2×D311yescharacteristic
107#400—61no—
108#398—61no—
109#397—62no—
110#391—61no—
111#389—61no—
112#380C2×D311yescharacteristic
113C4×C22⋊C4—31no—
114#339—62no—
115#328—62no—
116C24×C3C2×D411yescharacteristic
117C23×D3—21no—
118C22×C3×C4—22no—
119C22×C3×C4—42no—
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4#4111yescharacteristic
126C2×#67—42no—
127C2×#67—42no—
128C2×#67—41no—
129C2×#67—41no—
130C22×Dic3—21no—
131C22×Dic3—21no—
132C22×Dic3—21no—
133C22×Dic3—21no—
134C22×Dic3—21no—
135C22×Dic3—41no—
136C2×D12—21no—
137C2×D12—21no—
138C2×D12—42no—
139C2×Dic6—21no—
140C2×Dic6—21no—
141C2×Dic6—42no—
142C3×M16—42no—
143C3×M16—42no—
144C2×C3×C8—22no—
145C2×C3×C8—22no—
146C3×C4⋊C4—42no—
147C3×C22⋊C4—42no—
148#217—22no—
149#217—41no—
150#217—41no—
151#217—41no—
152#217—41no—
153#217—41no—
154(C2×D3)⋊C4—41no—
155(C2×D3)⋊C4—41no—
156C4⋊Dic3—21no—
157C4⋊Dic3—21no—
158C4⋊Dic3—21no—
159C4⋊Dic3—21no—
160C4⋊Dic3—21no—
161C4⋊Dic3—21no—
162C4⋊Dic3—21no—
163C4⋊Dic3—21no—
164C4⋊Dic3—82no—
165Dic3⋊C4—41no—
166Dic3⋊C4—41no—
167C4×Dic3—41no—
168C4×Dic3—41no—
169C4×Dic3—42no—
170#204—82no—
171C22×Q8—31no—
172C22×D4—31no—
173C22×D4—61no—
174C23×C4C22×D311yescharacteristic
175C2×SD16—62no—
176C2×M16—21no—
177C2×M16D1212yes—
178C22⋊Q8—61no—
179C22⋊Q8—61no—
180C22⋊Q8—61no—
181C22⋊Q8—61no—
182#121—61no—
183#121—61no—
184#121—61no—
185#121—61no—
186#117—61no—
187#117—61no—
188#117—61no—
189#117—61no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—31no—
192C2×C4⋊C4—31no—
193C2×C4⋊C4—31no—
194C2×C4⋊C4—31no—
195C2×C4⋊C4—61no—
196C2×C4⋊C4—62no—
197C2×C22⋊C4—61no—
198C2×C22⋊C4—61no—
199C2×C22⋊C4—61no—
200C2×C22⋊C4—61no—
201C2×C22⋊C4D1212yes—
202C2×C42—61no—
203C2×C42—61no—
204#107—62no—
205#107—62no—
206#106—62no—
207#103—62no—
208#102—62no—
209#102—62no—
210#102—62no—
211#99—124no—
212C22⋊C8—21no—
213C22⋊C8—21no—
214C22⋊C8—21no—
215C22⋊C8D1212yes—
216#95—21no—
217#95—62no—
218#95—62no—
219#95—62no—
220#95—61no—
221C23×C3—21no—
222C23×C3—21no—
223C23×C3—41no—
224C23×C3—41no—
225C23×C3#12011yescharacteristic
226C23×C3#12111yescharacteristic
227C23×C3#12711yescharacteristic
228C22×D3—41no—
229C22×D3—42no—
230C22×D3—81no—
231C22×D3—81no—
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—42no—
239C2×C3×C4—42no—
240C2×C3×C4—42no—
241C2×C3×C4#12011yescharacteristic
242C2×C3×C4#12011yescharacteristic
243C2×C3×C4#12111yescharacteristic
244C2×C3×C4#12111yescharacteristic
245#67—82no—
246#67—81no—
247#67—81no—
248C2×Dic3—42no—
249C2×Dic3—42no—
250C2×Dic3—42no—
251C2×Dic3—42no—
252C2×Dic3—41no—
253C2×Dic3—41no—
254C2×Dic3—41no—
255C2×Dic3—41no—
256C2×Dic3—41no—
257C2×Dic3—41no—
258C2×Dic3—41no—
259C2×Dic3—41no—
260C2×Dic3—41no—
261C2×Dic3—81no—
262C2×Dic3—81no—
263D12—42no—
264D12—42no—
265Dic6—42no—
266Dic6—81no—
267C3×C8—42no—
268C3×C8—42no—
269C24—61no—
270C24C2×D1211yescharacteristic
271C2×Q8—61no—
272C2×Q8—61no—
273C2×Q8—122no—
274C2×D4—61no—
275C2×D4—61no—
276C2×D4—122no—
277C2×D4—122no—
278C2×D4—122no—
279C2×D4—121no—
280C2×D4—121no—
281C22×C4—22no—
282C22×C4—42no—
283C22×C4—61no—
284C22×C4—61no—
285C22×C4—61no—
286C22×C4—61no—
287C22×C4—61no—
288C22×C4—121no—
289C22×C4C2×D1211yescharacteristic
290C22×C4C2×D1211yescharacteristic
291C22×C4D3×D411yescharacteristic
292C22×C4D3×D411yescharacteristic
293C22×C4D3×D411yescharacteristic
294C22×C4#23911yescharacteristic
295SD16—242no—
296M16—42no—
297M16—42no—
298C2×C8—22no—
299C2×C8—22no—
300C4⋊C4—42no—
301C4⋊C4—61no—
302C4⋊C4—61no—
303C4⋊C4—61no—
304C4⋊C4—61no—
305C4⋊C4—61no—
306C4⋊C4—61no—
307C4⋊C4—61no—
308C4⋊C4—61no—
309C4⋊C4—121no—
310C4⋊C4—121no—
311C4⋊C4—242no—
312C22⋊C4—42no—
313C22⋊C4—62no—
314C22⋊C4—121no—
315C22⋊C4—121no—
316C22⋊C4—121no—
317C22⋊C4—121no—
318C22⋊C4—121no—
319C22⋊C4—121no—
320C22⋊C4—121no—
321C42—122no—
322C42—121no—
323C42—121no—
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—21no—
327C22×C3—21no—
328C22×C3—21no—
329C22×C3—21no—
330C22×C3—21no—
331C22×C3—21no—
332C22×C3—81no—
333C22×C3—81no—
334C22×C3#39211yescharacteristic
335C22×C3#45311yescharacteristic
336C22×C3#45411yescharacteristic
337C2×D3—81no—
338C2×D3—81no—
339C2×D3—81no—
340C2×D3—162no—
341C3×C4—21no—
342C3×C4—21no—
343C3×C4—21no—
344C3×C4—82no—
345C3×C4#45812yes—
346Dic3—82no—
347Dic3—81no—
348Dic3—81no—
349Dic3—81no—
350Dic3—81no—
351Dic3—81no—
352C23—21no—
353C23—21no—
354C23—41no—
355C23—41no—
356C23—122no—
357C23—121no—
358C23—241no—
359C23—241no—
360C23#87411yescharacteristic
361C23#88211yescharacteristic
362C23#89511yescharacteristic
363Q8—122no—
364Q8—241no—
365D4—122no—
366D4—122no—
367D4—242no—
368D4—241no—
369D4—241no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—122no—
380C2×C4—122no—
381C2×C4—122no—
382C2×C4—122no—
383C2×C4—121no—
384C2×C4—121no—
385C2×C4—121no—
386C2×C4—121no—
387C2×C4—121no—
388C2×C4—121no—
389C2×C4—121no—
390C2×C4—121no—
391C2×C4—121no—
392C2×C4—241no—
393C2×C4—241no—
394C2×C4#88211yescharacteristic
395C2×C4#88211yescharacteristic
396C2×C4#89511yescharacteristic
397C2×C4#89511yescharacteristic
398C8—42no—
399C8—42no—
400C2×C3—21no—
401C2×C3—21no—
402C2×C3—41no—
403C2×C3—41no—
404C2×C3#200611yescharacteristic
405C2×C3#200911yescharacteristic
406C2×C3#202211yescharacteristic
407D3—161no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—81no—
417C22—81no—
418C22—241no—
419C22—241no—
420C22—241no—
421C22—482no—
422C22#467111yescharacteristic
423C22#474711yescharacteristic
424C22#475311yescharacteristic
425C4—21no—
426C4—21no—
427C4—21no—
428C4—82no—
429C4—242no—
430C4—241no—
431C4—241no—
432C4—241no—
433C4—241no—
434C4—241no—
435C4#474212yes—
436C3#993311yescharacteristic, complemented, Sylow
437C2—21no—
438C2—21no—
439C2—41no—
440C2—41no—
441C2—481no—
442C2#6896611yescharacteristic
443C2#6900811yescharacteristic
444C2#6903111yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1305C2
socleC22×C3#392
prime radicalC2×#930C22
hypercenter#1305D3
hypocenterC3#9933
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1305

Complements

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

NH ≅ G/Ntype
3. #73951C2semidirect
4. #69980C2semidirect
5. #69975C2semidirect
7. C3×#1305C2semidirect
40. #1305D3semidirect
436. C3#9933semidirect

Maximal subgroups

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

subgroupindex
#786422
#786242
#739512
#699802
#699752
#695282
C3×#13052
#99333