Finite Group Databasebeta

← #10693140

Subgroups — #10693140

Order 768.

682 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 682 conjugacy classes of subgroups — too many to draw. Only its 95 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#78634C211yesmaximal, characteristic
2#74017C211yesmaximal, characteristic
3#69990C211yesmaximal, characteristic, complemented
4#69990C211yesmaximal, characteristic, complemented
5#69968C211yesmaximal, characteristic
6C3×#1305C211yesmaximal, characteristic, complemented
7#67626C211yesmaximal, characteristic, complemented
8#9506—31nomaximal, Sylow
9C23×Dic6C2211yescharacteristic
10C2×C4×#67C2211yescharacteristic
11C2×#897—21no—
12C3×#407C2211yescharacteristic
13C3×#407C2211yescharacteristic
14C3×#380C2211yescharacteristic
15#5207C2211yescharacteristic
16#5207C2211yescharacteristic
17C2×#824—22no—
18C2×#824—22no—
19#5128—22no—
20#4973—21no—
21#4958C412yes—
22#4950C412yes—
23#4729—22no—
24#4726—21no—
25#4718—21no—
26#4673—21no—
27#4656—21no—
28#4469—21no—
29#4468—21no—
30#2975—31no—
31#2409—31no—
32#1784—31no—
33#1784—31no—
34#1465—31no—
35#1305—31no—
36#1305D311yescharacteristic, complemented
37C23×C3×C4C2311yescharacteristic
38C22×#67C2×C411yescharacteristic
39C23×Dic3C2×C412yes—
40C22×C4×D3D411yescharacteristic
41C22×Dic6—21no—
42C22×Dic6—21no—
43C22×Dic6—21no—
44C22×Dic6—21no—
45C22×Dic6—21no—
46C22×Dic6—21no—
47C22×Dic6—44no—
48C2×C3×M16—21no—
49C2×C3×M16D412yes—
50C2×C3×C4⋊C4—21no—
51C2×C3×C22⋊C4C2×C412yes—
52C2×#217C2×C411yescharacteristic
53C4×#67—21no—
54C4×#67—21no—
55C4×#67—21no—
56C4×#67—21no—
57C2×(C2×D3)⋊C4—22no—
58C2×(C2×D3)⋊C4D411yescharacteristic
59C2×C4⋊Dic3—22no—
60C2×Dic3⋊C4D411yescharacteristic
61C2×C4×Dic3D411yescharacteristic
62C2×#208—21no—
63C2×#208D412yes—
64C2×ΓL(1,𝔽25)—22no—
65#897—44no—
66C3×C22⋊C8—21no—
67C3×C22⋊C8—21no—
68C3×C22⋊C8—21no—
69C3×C22⋊C8—21no—
70C3×#95—21no—
71C3⋊#95—22no—
72C3⋊#95—22no—
73#830—21no—
74#830—21no—
75#830—21no—
76#830—21no—
77#824—44no—
78#824—44no—
79(C2×D3)⋊C8—22no—
80Dic3⋊C8—22no—
81C8×Dic3—22no—
82C23×Q8—31no—
83C2×C22⋊Q8—61no—
84C2×C4×D4—31no—
85#436—62no—
86#436—62no—
87#431—61no—
88C2×#101—62no—
89C2×#101—62no—
90#407—31no—
91#407—31no—
92#407—61no—
93#407C2×D311yescharacteristic
94#407C2×D311yescharacteristic
95#395—61no—
96#393—61no—
97#391—61no—
98#386—32no—
99#385—32no—
100#380C2×D311yescharacteristic
101#324—61no—
102#323—61no—
103C24×C3C22×C411yescharacteristic
104C23×D3C22⋊C411yescharacteristic
105C22×C3×C4—22no—
106C22×C3×C4—42no—
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4#4111yescharacteristic
112C22×C3×C4#4111yescharacteristic
113C2×#67—21no—
114C2×#67—21no—
115C2×#67—22no—
116C2×#67—41no—
117C22×Dic3—22no—
118C22×Dic3—22no—
119C22×Dic3—22no—
120C22×Dic3—44no—
121C22×Dic3C22⋊C411yescharacteristic
122C22×Dic3C22⋊C411yescharacteristic
123C22×Dic3C22⋊C411yescharacteristic
124C2×C4×D3—21no—
125C2×C4×D3—21no—
126C2×C4×D3—42no—
127C2×Dic6—24no—
128C2×Dic6—24no—
129C2×Dic6—24no—
130C2×Dic6—44no—
131C2×Dic6—44no—
132C2×Dic6—44no—
133C2×Dic6—42no—
134C2×Dic6—42no—
135C2×Dic6—42no—
136C2×Dic6—41no—
137C2×Dic6—41no—
138C2×Dic6—41no—
139C2×Dic6—41no—
140C3×M16—42no—
141C3×M16—42no—
142C2×C3×C8—22no—
143C2×C3×C8—41no—
144C3×C4⋊C4—42no—
145C3×C22⋊C4—42no—
146#217—21no—
147#217—21no—
148(C2×D3)⋊C4—21no—
149(C2×D3)⋊C4—21no—
150(C2×D3)⋊C4—44no—
151C4⋊Dic3—44no—
152Dic3⋊C4—21no—
153Dic3⋊C4—21no—
154C4×Dic3—21no—
155C4×Dic3—21no—
156#208—42no—
157#208—42no—
158C2×C3⋊C8—22no—
159C2×C3⋊C8—41no—
160ΓL(1,𝔽25)—82no—
161C22×Q8—61no—
162C22×Q8—61no—
163C22×Q8—61no—
164C22×Q8—61no—
165C22×Q8—61no—
166C22×Q8—61no—
167C22×Q8—124no—
168C22×D4—31no—
169C23×C4—31no—
170C23×C4—32no—
171C23×C4C22×D311yescharacteristic
172C2×M16—21no—
173C2×M16—32no—
174C2×M16—62no—
175C2×M16—61no—
176C2×M16D1212yes—
177C22⋊Q8—124no—
178C4×D4—61no—
179C4×D4—61no—
180C4×D4—61no—
181C4×D4—61no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—31no—
184C2×C4⋊C4—62no—
185C2×C22⋊C4—31no—
186C2×C22⋊C4—31no—
187C2×C22⋊C4—62no—
188C2×C22⋊C4C4×D312yes—
189C2×C42—31no—
190C4⋊C8—62no—
191#101—124no—
192#101—124no—
193C22⋊C8—21no—
194C22⋊C8—21no—
195C22⋊C8—21no—
196C22⋊C8—21no—
197C22⋊C8—62no—
198C22⋊C8—61no—
199C22⋊C8—61no—
200C22⋊C8—61no—
201C22⋊C8—61no—
202C4×C8—62no—
203#95—21no—
204#95—62no—
205#95—62no—
206C23×C3—21no—
207C23×C3—21no—
208C23×C3—41no—
209C23×C3—41no—
210C23×C3C2×C22⋊C411yescharacteristic
211C23×C3C4×D411yescharacteristic
212C23×C3C4×D411yescharacteristic
213C22×D3—21no—
214C22×D3—21no—
215C22×D3—21no—
216C22×D3—41no—
217C22×D3—41no—
218C2×C3×C4—22no—
219C2×C3×C4—22no—
220C2×C3×C4—22no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—42no—
225C2×C3×C4—42no—
226C2×C3×C4—42no—
227C2×C3×C4#12011yescharacteristic
228C2×C3×C4#12111yescharacteristic
229C2×C3×C4#12111yescharacteristic
230C2×C3×C4#12311yescharacteristic
231#67—42no—
232#67—42no—
233C2×Dic3—21no—
234C2×Dic3—21no—
235C2×Dic3—24no—
236C2×Dic3—24no—
237C2×Dic3—24no—
238C2×Dic3—21no—
239C2×Dic3—21no—
240C2×Dic3—42no—
241C2×Dic3—42no—
242C2×Dic3—42no—
243C2×Dic3—42no—
244C2×Dic3—41no—
245C2×Dic3—41no—
246C2×Dic3#10112yescomplemented
247C4×D3—42no—
248C4×D3—81no—
249Dic6—44no—
250Dic6—44no—
251Dic6—44no—
252Dic6—44no—
253C3×C8—42no—
254C3×C8—81no—
255C3⋊C8—42no—
256C3⋊C8—81no—
257C24—31no—
258C24C2×C4×D311yescharacteristic
259C2×Q8—64no—
260C2×Q8—64no—
261C2×Q8—64no—
262C2×Q8—122no—
263C2×Q8—122no—
264C2×Q8—122no—
265C2×Q8—124no—
266C2×Q8—124no—
267C2×Q8—124no—
268C2×Q8—121no—
269C2×Q8—121no—
270C2×Q8—121no—
271C2×Q8—121no—
272C2×D4—62no—
273C2×D4—61no—
274C2×D4—61no—
275C2×D4—121no—
276C22×C4—22no—
277C22×C4—31no—
278C22×C4—31no—
279C22×C4—31no—
280C22×C4—42no—
281C22×C4—62no—
282C22×C4—62no—
283C22×C4—62no—
284C22×C4—61no—
285C22×C4—61no—
286C22×C4—122no—
287C22×C4—124no—
288C22×C4C2×D1211yescharacteristic
289C22×C4#23511yescharacteristic
290C22×C4D3×D411yescharacteristic
291C22×C4D3×D411yescharacteristic
292C22×C4D3×D411yescharacteristic
293C22×C4#23911yescharacteristic
294M16—42no—
295M16—42no—
296M16—122no—
297M16—122no—
298M16—242no—
299C2×C8—22no—
300C2×C8—41no—
301C2×C8—62no—
302C2×C8—121no—
303C4⋊C4—42no—
304C4⋊C4—61no—
305C4⋊C4—61no—
306C4⋊C4—124no—
307C22⋊C4—42no—
308C22⋊C4—61no—
309C22⋊C4—61no—
310C22⋊C4—61no—
311C22⋊C4—61no—
312C22⋊C4—124no—
313C42—61no—
314C42—61no—
315C22×C3—21no—
316C22×C3—21no—
317C22×C3—21no—
318C22×C3—21no—
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—42no—
324C22×C3—42no—
325C22×C3#38611yescharacteristic
326C22×C3C2×#10111yescharacteristic
327C22×C3#41311yescharacteristic
328C2×D3—42no—
329C2×D3—41no—
330C2×D3—41no—
331C2×D3—41no—
332C2×D3—81no—
333C3×C4—21no—
334C3×C4—21no—
335C3×C4—21no—
336C3×C4—82no—
337C3×C4B(4,𝔽2)12yes—
338Dic3—22no—
339Dic3—44no—
340Dic3—41no—
341Dic3—41no—
342C23—21no—
343C23—21no—
344C23—41no—
345C23—41no—
346C23—61no—
347C23—61no—
348C23—61no—
349C23—121no—
350C23—121no—
351C23C4×D1211yescharacteristic
352C23D3×C22⋊C411yescharacteristic
353C23#89311yescharacteristic
354Q8—124no—
355Q8—124no—
356Q8—124no—
357Q8—124no—
358D4—122no—
359D4—122no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—32no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—61no—
371C2×C4—61no—
372C2×C4—64no—
373C2×C4—64no—
374C2×C4—64no—
375C2×C4—61no—
376C2×C4—61no—
377C2×C4—122no—
378C2×C4—122no—
379C2×C4—122no—
380C2×C4—122no—
381C2×C4—122no—
382C2×C4—121no—
383C2×C4—121no—
384C2×C4—241no—
385C2×C4#88211yescharacteristic
386C2×C4#88411yescharacteristic
387C2×C4#89411yescharacteristic
388C2×C4#89511yescharacteristic
389C8—42no—
390C8—81no—
391C8—122no—
392C8—241no—
393C2×C3—21no—
394C2×C3—21no—
395C2×C3—41no—
396C2×C3—41no—
397C2×C3#188911yescharacteristic
398C2×C3#190111yescharacteristic
399C2×C3#191511yescharacteristic
400D3—81no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—42no—
410C22—42no—
411C22—122no—
412C22—121no—
413C22—121no—
414C22—121no—
415C22—241no—
416C22#466511yescharacteristic
417C22#474511yescharacteristic
418C22D3×#10111yescharacteristic
419C4—21no—
420C4—21no—
421C4—21no—
422C4—62no—
423C4—82no—
424C4—124no—
425C4—121no—
426C4—121no—
427C4#474112yes—
428C3#950611yescharacteristic, complemented, Sylow
429C2—21no—
430C2—21no—
431C2—41no—
432C2—41no—
433C2—241no—
434C2#6896211yescharacteristic
435C2#6899311yescharacteristic
436C2#6904111yescharacteristic
subgroupG/H
centerC22#4665
commutatorC24×C3C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1305C2
socleC22×C3#386
prime radicalC22×#67C2×C4
cosocleC23×C3×C4C23
hypercenter#1305D3
hypocenterC3#9506
LCSG ▷ C24×C3 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1305

Complements

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

NH ≅ G/Ntype
3. #69990C2semidirect
4. #69990C2semidirect
6. C3×#1305C2semidirect
7. #67626C2semidirect
36. #1305D3semidirect
246. C2×Dic3#101semidirect
428. C3#9506semidirect

Maximal subgroups

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

subgroupindex
#786342
#740172
#699902
#699682
C3×#13052
#676262
#95063