Finite Group Databasebeta

← #10698102

Subgroups — #10698102

Order 768.

780 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 87 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#82760C211yesmaximal, characteristic, complemented
2#82738C211yesmaximal, characteristic
3#73491C211yesmaximal, characteristic
4C3×#1508C211yesmaximal, characteristic, complemented
5#69990C211yesmaximal, characteristic, complemented
6#69977C211yesmaximal, characteristic
7#67618C211yesmaximal, characteristic, complemented
8#10330—31nomaximal, Sylow
9C23×Dic6C2211yescharacteristic
10C2×C3×C22⋊Q8C2211yescharacteristic
11C2×#944—21no—
12C2×#930C2211yescharacteristic
13C2×#875—21no—
14C3×#407C2211yescharacteristic
15C3×C4×C22⋊C4C2211yescharacteristic
16#5227—21no—
17C4×#217C2211yescharacteristic
18#5207C2211yescharacteristic
19C2×#824—22no—
20#5049—22no—
21#5046—21no—
22#5044—21no—
23#4979—22no—
24#4973—21no—
25#4967—21no—
26#4958—22no—
27#4946—22no—
28#4929—21no—
29#4673—22no—
30C3×#328—22no—
31#4470—22no—
32#3193—31no—
33#2436—31no—
34#2434—31no—
35#1784—31no—
36#1508D311yescharacteristic, complemented
37#1507—31no—
38#1297—31no—
39C22×C3×Q8D411yescharacteristic
40C23×C3×C4C2311yescharacteristic
41C22×#67—21no—
42C23×Dic3D412yes—
43C22×D12D411yescharacteristic
44C22×Dic6—21no—
45C22×Dic6—21no—
46C22×Dic6—21no—
47C22×Dic6—21no—
48C22×Dic6—21no—
49C22×Dic6—21no—
50C22×Dic6—44no—
51C2×C3×M16—21no—
52C3×C22⋊Q8—21no—
53C3×C22⋊Q8—21no—
54C3×C22⋊Q8—21no—
55C3×C22⋊Q8—21no—
56C2×C3×C4⋊C4—21no—
57C2×C3×C4⋊C4D411yescharacteristic
58C2×C3×C22⋊C4—21no—
59C2×C3×C22⋊C4—21no—
60C2×C3×C42—21no—
61C2×#217—21no—
62#944—44no—
63C2×#215—22no—
64#930—21no—
65#930—21no—
66#930—21no—
67#930—21no—
68C2×(C2×D3)⋊C4—24no—
69C2×(C2×D3)⋊C4—21no—
70C2×C4⋊Dic3D411yescharacteristic
71C2×Dic3⋊C4—24no—
72C2×C4×Dic3—21no—
73C2×#208D412yes—
74#875—44no—
75C3×C22⋊C8—21no—
76C3×C22⋊C8—21no—
77C3×#95D412yes—
78#835—22no—
79C3⋊#95—24no—
80C3⋊#95—24no—
81C3⋊#95D412yes—
82#830—21no—
83#830—21no—
84#824—44no—
85#809—22no—
86#807—22no—
87C23×Q8—31no—
88C2×C4⋊Q8—61no—
89C2×#124—61no—
90C2×C22⋊Q8C2×D311yescharacteristic
91C2×#121—31no—
92#484—61no—
93#480—61no—
94#468—62no—
95#450—61no—
96#448—61no—
97C2×#101—62no—
98#407—31no—
99#407C2×D311yescharacteristic
100#395—62no—
101#393—62no—
102#391—61no—
103#391—61no—
104#386—62no—
105#385—62no—
106C4×C22⋊C4—31no—
107C4×C22⋊C4C2×D311yescharacteristic
108#328—22no—
109#327—62no—
110C24×C3C2×D411yescharacteristic
111C23×D3—21no—
112C2×C3×Q8—21no—
113C2×C3×Q8—21no—
114C2×C3×Q8—42no—
115C22×C3×C4—21no—
116C22×C3×C4—22no—
117C22×C3×C4—21no—
118C22×C3×C4—42no—
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4#4111yescharacteristic
125C2×#67—44no—
126C2×#67—41no—
127C2×#67—41no—
128C22×Dic3—22no—
129C22×Dic3—22no—
130C22×Dic3—22no—
131C22×Dic3—22no—
132C22×Dic3—21no—
133C22×Dic3—44no—
134C2×D12—21no—
135C2×D12—21no—
136C2×D12—42no—
137C2×Dic6—24no—
138C2×Dic6—24no—
139C2×Dic6—24no—
140C2×Dic6—44no—
141C2×Dic6—44no—
142C2×Dic6—44no—
143C2×Dic6—42no—
144C2×Dic6—42no—
145C2×Dic6—42no—
146C2×Dic6—41no—
147C2×Dic6—41no—
148C2×Dic6—41no—
149C2×Dic6—41no—
150C3×M16—42no—
151C2×C3×C8—41no—
152C3×C4⋊C4—21no—
153C3×C4⋊C4—21no—
154C3×C4⋊C4—41no—
155C3×C4⋊C4—41no—
156C3×C22⋊C4—42no—
157C3×C22⋊C4—41no—
158C3×C22⋊C4—41no—
159C3×C42—42no—
160#217—24no—
161#215—82no—
162(C2×D3)⋊C4—48no—
163(C2×D3)⋊C4—41no—
164(C2×D3)⋊C4—41no—
165C4⋊Dic3—21no—
166C4⋊Dic3—21no—
167Dic3⋊C4—48no—
168C4×Dic3—42no—
169#208—42no—
170C2×C3⋊C8—22no—
171C22×Q8—61no—
172C22×Q8—61no—
173C22×Q8—61no—
174C22×Q8—61no—
175C22×Q8—61no—
176C22×Q8—61no—
177C22×Q8—124no—
178C22×Q8D1211yescharacteristic
179C22×D4—31no—
180C22×D4—61no—
181C23×C4—32no—
182C23×C4C22×D311yescharacteristic
183C2×SD16—62no—
184C2×M16—21no—
185C2×M16—32no—
186C4⋊Q8—124no—
187#124—124no—
188C22⋊Q8—21no—
189C22⋊Q8—21no—
190C22⋊Q8—21no—
191C22⋊Q8—21no—
192#121—61no—
193#121—61no—
194#121—61no—
195#121—61no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—31no—
198C2×C4⋊C4—64no—
199C2×C4⋊C4D1211yescharacteristic
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—61no—
203C2×C22⋊C4—64no—
204C2×C22⋊C4—61no—
205C2×C42—21no—
206C2×C42—61no—
207#107—62no—
208#103—62no—
209#102—62no—
210#101—124no—
211C22⋊C8—21no—
212C22⋊C8—21no—
213C22⋊C8—61no—
214C22⋊C8—61no—
215#95—32no—
216#95—64no—
217#95—64no—
218#95#6712yes—
219C23×C3—22no—
220C23×C3—22no—
221C23×C3—41no—
222C23×C3#12011yescharacteristic
223C23×C3#12011yescharacteristic
224C23×C3#12111yescharacteristic
225C22×D3—42no—
226C22×D3—42no—
227C22×D3—41no—
228C22×D3—42no—
229C3×Q8—42no—
230C3×Q8—81no—
231C2×C3×C4—22no—
232C2×C3×C4—22no—
233C2×C3×C4—22no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4—42no—
238C2×C3×C4—42no—
239C2×C3×C4—42no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—42no—
243C2×C3×C4—42no—
244C2×C3×C4#12011yescharacteristic
245C2×C3×C4#12111yescharacteristic
246C2×C3×C4#12111yescharacteristic
247C2×C3×C4#12711yescharacteristic
248#67—82no—
249#67—82no—
250C2×Dic3—24no—
251C2×Dic3—24no—
252C2×Dic3—24no—
253C2×Dic3—24no—
254C2×Dic3—42no—
255C2×Dic3—42no—
256C2×Dic3—42no—
257C2×Dic3—42no—
258C2×Dic3—42no—
259C2×Dic3—42no—
260C2×Dic3—42no—
261C2×Dic3—41no—
262D12—42no—
263D12—81no—
264Dic6—44no—
265Dic6—44no—
266Dic6—44no—
267Dic6—44no—
268C3×C8—81no—
269C3⋊C8—42no—
270C24—61no—
271C24D3×D411yescharacteristic
272C2×Q8—21no—
273C2×Q8—21no—
274C2×Q8—42no—
275C2×Q8—64no—
276C2×Q8—64no—
277C2×Q8—64no—
278C2×Q8—122no—
279C2×Q8—122no—
280C2×Q8—122no—
281C2×Q8—124no—
282C2×Q8—124no—
283C2×Q8—124no—
284C2×Q8—121no—
285C2×Q8—121no—
286C2×Q8—121no—
287C2×Q8—121no—
288C2×D4—61no—
289C2×D4—61no—
290C2×D4—122no—
291C2×D4—124no—
292C2×D4—121no—
293C2×D4—121no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—42no—
298C22×C4—62no—
299C22×C4—62no—
300C22×C4—62no—
301C22×C4—62no—
302C22×C4—61no—
303C22×C4—124no—
304C22×C4C2×D1211yescharacteristic
305C22×C4#23511yescharacteristic
306C22×C4D3×D411yescharacteristic
307C22×C4D3×D411yescharacteristic
308C22×C4D3×D411yescharacteristic
309C22×C4C2×#6711yescharacteristic
310SD16—242no—
311M16—42no—
312M16—122no—
313C2×C8—41no—
314C2×C8—62no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—61no—
320C4⋊C4—61no—
321C4⋊C4—128no—
322C22⋊C4—42no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—64no—
326C22⋊C4—128no—
327C22⋊C4—121no—
328C22⋊C4—121no—
329C42—42no—
330C42—122no—
331C22×C3—21no—
332C22×C3—21no—
333C22×C3—21no—
334C22×C3—21no—
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—21no—
338C22×C3—44no—
339C22×C3#39211yescharacteristic
340C22×C3#45412yes—
341C22×C3#45612yes—
342C2×D3—84no—
343C2×D3—81no—
344C2×D3—81no—
345C2×D3—81no—
346C3×C4—21no—
347C3×C4—21no—
348C3×C4—21no—
349C3×C4—82no—
350C3×C4—81no—
351C3×C4—81no—
352C3×C4B(4,𝔽2)12yes—
353Dic3—44no—
354Dic3—44no—
355Dic3—81no—
356C23—22no—
357C23—22no—
358C23—41no—
359C23—122no—
360C23—122no—
361C23—122no—
362C23—121no—
363C23#88211yescharacteristic
364C23#88411yescharacteristic
365C23#93711yescharacteristic
366Q8—42no—
367Q8—81no—
368Q8—124no—
369Q8—124no—
370Q8—124no—
371Q8—124no—
372D4—122no—
373D4—241no—
374D4—242no—
375D4—242no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—64no—
390C2×C4—64no—
391C2×C4—64no—
392C2×C4—64no—
393C2×C4—122no—
394C2×C4—122no—
395C2×C4—122no—
396C2×C4—122no—
397C2×C4—122no—
398C2×C4—122no—
399C2×C4—122no—
400C2×C4—121no—
401C2×C4#88211yescharacteristic
402C2×C4#88411yescharacteristic
403C2×C4#93011yescharacteristic
404C2×C4#94011yescharacteristic
405C8—81no—
406C8—122no—
407C2×C3—22no—
408C2×C3—22no—
409C2×C3—41no—
410C2×C3#200311yescharacteristic
411C2×C3#200511yescharacteristic
412C2×C3#202111yescharacteristic
413D3—161no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—44no—
422C22—244no—
423C22—241no—
424C22—241no—
425C22—241no—
426C22#482712yes—
427C22#496011yescharacteristic
428C22#520512yes—
429C4—21no—
430C4—21no—
431C4—21no—
432C4—82no—
433C4—81no—
434C4—81no—
435C4—124no—
436C4—124no—
437C4—241no—
438C4#474112yes—
439C3#1033011yescharacteristic, complemented, Sylow
440C2—22no—
441C2—22no—
442C2—41no—
443C2—481no—
444C2#7003711yescharacteristic
445C2#7034611yescharacteristic
446C2#7038811yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1508C2
socleC22×C3#392
prime radicalC2×#930C22
hypercenter#1508D3
hypocenterC3#10330
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1508

Complements

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

NH ≅ G/Ntype
1. #82760C2semidirect
4. C3×#1508C2semidirect
5. #69990C2semidirect
7. #67618C2semidirect
36. #1508D3semidirect
439. C3#10330semidirect

Maximal subgroups

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

subgroupindex
#827602
#827382
#734912
C3×#15082
#699902
#699772
#676182
#103303