Finite Group Databasebeta

← #10696032

Subgroups — #10696032

Order 768.

598 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C3×#3100C211yesmaximal, characteristic
2#83315C211yesmaximal, characteristic, complemented
3#82348C211yesmaximal, characteristic, complemented
4#71816C211yesmaximal, characteristic, complemented
5#71598C211yesmaximal, characteristic, complemented
6#70770C211yesmaximal, characteristic, complemented
7#67737C211yesmaximal, characteristic, complemented
8#10013—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×C3×C4×D4C2211yescharacteristic
11C2×C4×Dic6C2211yescharacteristic
12C3×#480—21no—
13C3×#474—21no—
14C3×#459—21no—
15C3×#447—21no—
16C3×C4×D8—22no—
17C3×#431C2211yescharacteristic
18C2×#837—22no—
19#5222—21no—
20C4×#217—22no—
21#5094—21no—
22#5075—21no—
23#5040—21no—
24#5034—21no—
25C4×#214—22no—
26#4985—21no—
27#4953—21no—
28#4947—21no—
29#4932C2211yescharacteristic
30#4925C2211yescharacteristic
31#4924C2211yescharacteristic
32D4⋊(C3⋊C8)—22no—
33#4456—22no—
34#3103—31no—
35#3100D311yescharacteristic
36#2398—31no—
37#1963—31no—
38#1487—31no—
39#1486—31no—
40#1274—31no—
41C2×C3×#41—21no—
42C22×C3×D4D411yescharacteristic
43C23×C3×C4—21no—
44C22×Dic6D411yescharacteristic
45C2×C3×D8—22no—
46C3×C4⋊Q8D411yescharacteristic
47C3×#127D411yescharacteristic
48C3×#124—21no—
49C3×#121—21no—
50C3×#121—21no—
51C3×C4×D4—21no—
52C3×C4×D4—21no—
53C3×C4×D4—21no—
54C3×C4×D4—21no—
55C3×C4×D4—21no—
56C3×C4×D4—21no—
57C2×C3×C4⋊C4D411yescharacteristic
58C2×C3×C22⋊C4—21no—
59C2×C3×C42C2311yescharacteristic
60C2×#217—22no—
61C2×#217—22no—
62C2×#214—22no—
63C2×C4⋊Dic3D411yescharacteristic
64C2×Dic3⋊C4—21no—
65C2×C4×Dic3—21no—
66C2×C4×Dic3D412yes—
67C2×#208—21no—
68C4×Dic6—21no—
69C4×Dic6—21no—
70C4×Dic6—21no—
71C4×Dic6—21no—
72C3×#107—22no—
73C3×C4⋊C8—21no—
74C3×#102—22no—
75C3×#102—22no—
76C3×C22⋊C8—21no—
77C3×C4×C8D412yes—
78#837—44no—
79#832—22no—
80C3⋊#95—22no—
81C3⋊#95—22no—
82C3⋊#95—21no—
83C3⋊#95—21no—
84#830—21no—
85#810—22no—
86#808—22no—
87C4⋊(C3⋊C8)—21no—
88C4⋊(C3⋊C8)—21no—
89C4⋊(C3⋊C8)—21no—
90C4×C3⋊C8D412yes—
91#532C2×D311yescharacteristic
92C2×C4×Q8—31no—
93C2×C4×D4C2×D311yescharacteristic
94#481—61no—
95#480—21no—
96#475—61no—
97#474—21no—
98#460—61no—
99#459—21no—
100#448—61no—
101#447—21no—
102C4×SD16—62no—
103C4×D8—22no—
104#431—31no—
105#431C2×D311yescharacteristic
106#423—31no—
107C2×#104—62no—
108#400—61no—
109#397—61no—
110#394—61no—
111#393—61no—
112C4×C22⋊C4—62no—
113#376—31no—
114Q8⋊C8—62no—
115D4⋊C8—62no—
116C24×C3—21no—
117C3×#41—42no—
118C3×#41—42no—
119C2×C3×Q8—21no—
120C2×C3×D4—21no—
121C2×C3×D4—21no—
122C2×C3×D4—21no—
123C2×C3×D4—21no—
124C2×C3×D4—21no—
125C2×C3×D4—42no—
126C2×C3×D4—41no—
127C22×C3×C4—21no—
128C22×C3×C4—41no—
129C22×C3×C4—41no—
130C22×C3×C4—41no—
131C22×C3×C4—42no—
132C22×C3×C4—42no—
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×Dic3—21no—
137C22×Dic3—22no—
138C22×Dic3—21no—
139C2×Dic6—21no—
140C2×Dic6—21no—
141C2×Dic6—42no—
142C3×D8—44no—
143C2×C3×C8—22no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—21no—
147C3×C4⋊C4—21no—
148C3×C22⋊C4—41no—
149C3×C22⋊C4—41no—
150C3×C22⋊C4—41no—
151C3×C22⋊C4—41no—
152C3×C42C2×D411yescharacteristic
153C3×C42C2×D411yescharacteristic
154C3×C42C2×D411yescharacteristic
155C3×C42#4111yescharacteristic
156#217—44no—
157#217—44no—
158#214—44no—
159C4⋊Dic3—21no—
160C4⋊Dic3—21no—
161Dic3⋊C4—41no—
162Dic3⋊C4—41no—
163C4×Dic3—22no—
164C4×Dic3—22no—
165C4×Dic3—41no—
166C4×Dic3—41no—
167#208—42no—
168C2×C3⋊C8—22no—
169C2×C3⋊C8—41no—
170C2×#41—21no—
171C22×Q8—31no—
172C22×D4#6711yescharacteristic
173C23×C4—21no—
174C2×SD16—62no—
175C2×D8—22no—
176C2×M16—61no—
177C4⋊Q8#6711yescharacteristic
178#127#6711yescharacteristic
179#124—21no—
180#121—21no—
181#121—21no—
182C4×Q8—61no—
183C4×Q8—61no—
184C4×Q8—61no—
185C4×Q8—61no—
186C4×D4—21no—
187C4×D4—21no—
188C4×D4—21no—
189C4×D4—21no—
190C4×D4—21no—
191C4×D4—21no—
192C2×C4⋊C4—31no—
193C2×C4⋊C4—61no—
194C2×C4⋊C4#6711yescharacteristic
195C2×C22⋊C4—21no—
196C2×C22⋊C4—62no—
197C2×C22⋊C4—62no—
198C2×C42—32no—
199C2×C42—61no—
200C2×C42C22×D311yescharacteristic
201#107—22no—
202#106—62no—
203C4⋊C8—21no—
204C4⋊C8—61no—
205C4⋊C8—61no—
206C4⋊C8—61no—
207#104—124no—
208#103—62no—
209#102—22no—
210#102—22no—
211#102—62no—
212C22⋊C8—21no—
213C22⋊C8—61no—
214C4×C8—32no—
215C4×C8#6712yes—
216#95—62no—
217#95—62no—
218#95—61no—
219#95—61no—
220C23×C3—22no—
221C23×C3—42no—
222C23×C3—41no—
223C23×C3—41no—
224C23×C3—42no—
225C23×C3—41no—
226C23×C3#12011yescharacteristic
227C3×Q8—42no—
228C3×D4—22no—
229C3×D4—22no—
230C3×D4—41no—
231C3×D4—41no—
232C3×D4—42no—
233C3×D4—42no—
234C3×D4—42no—
235C3×D4—82no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—42no—
241C2×C3×C4—82no—
242C2×C3×C4—81no—
243C2×C3×C4—81no—
244C2×C3×C4—81no—
245C2×C3×C4—81no—
246C2×C3×C4#12011yescharacteristic
247C2×C3×C4#12011yescharacteristic
248C2×C3×C4#12111yescharacteristic
249C2×C3×C4#12111yescharacteristic
250C2×C3×C4#12111yescharacteristic
251C2×C3×C4#12711yescharacteristic
252C2×C3×C4#13512yes—
253C2×C3×C4AGL(1,ℤ/8)12yes—
254C2×Dic3—22no—
255C2×Dic3—41no—
256C2×Dic3—41no—
257C2×Dic3—42no—
258C2×Dic3—42no—
259C2×Dic3—42no—
260C2×Dic3—41no—
261C2×Dic3—42no—
262Dic6—22no—
263Dic6—22no—
264Dic6—41no—
265Dic6—41no—
266C3×C8—42no—
267C3⋊C8—42no—
268C3⋊C8—81no—
269C24—21no—
270#41—42no—
271#41—42no—
272C2×Q8—21no—
273C2×Q8—61no—
274C2×Q8—61no—
275C2×Q8—122no—
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—21no—
281C2×D4—42no—
282C2×D4—41no—
283C22×C4—21no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—61no—
290C22×C4—62no—
291C22×C4—61no—
292C22×C4D3×D411yescharacteristic
293C22×C4D3×D411yescharacteristic
294C22×C4C2×#6711yescharacteristic
295SD16—124no—
296D8—44no—
297M16—122no—
298C2×C8—22no—
299C2×C8—62no—
300C2×C8—121no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—61no—
306C4⋊C4—61no—
307C4⋊C4—121no—
308C4⋊C4—121no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—124no—
314C22⋊C4—124no—
315C42—62no—
316C42—62no—
317C42—121no—
318C42—121no—
319C42D3×D411yescharacteristic
320C42#23711yescharacteristic
321C42C2×#6711yescharacteristic
322C42C2×#6711yescharacteristic
323C22×C3—21no—
324C22×C3—21no—
325C22×C3—44no—
326C22×C3—42no—
327C22×C3—82no—
328C22×C3—81no—
329C22×C3—81no—
330C22×C3—81no—
331C22×C3—81no—
332C22×C3—81no—
333C22×C3#39211yescharacteristic
334C22×C3#44912yes—
335C3×C4—21no—
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—81no—
340C3×C4—81no—
341C3×C4#46312yes—
342C3×C4#49511yescharacteristic
343C3×C4#49611yescharacteristic
344Dic3—42no—
345Dic3—82no—
346Dic3—81no—
347C23—22no—
348C23—42no—
349C23—41no—
350C23—42no—
351C23—41no—
352C23—41no—
353C23#93711yescharacteristic
354Q8—42no—
355Q8—62no—
356Q8—62no—
357Q8—121no—
358Q8—121no—
359D4—22no—
360D4—22no—
361D4—41no—
362D4—41no—
363D4—42no—
364D4—42no—
365D4—42no—
366D4—82no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—62no—
373C2×C4—82no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—121no—
379C2×C4—121no—
380C2×C4—122no—
381C2×C4—122no—
382C2×C4—122no—
383C2×C4—122no—
384C2×C4—121no—
385C2×C4#91112yes—
386C2×C4#91212yes—
387C2×C4#93711yescharacteristic
388C2×C4#93811yescharacteristic
389C2×C4#93911yescharacteristic
390C2×C4#93911yescharacteristic
391C2×C4#94011yescharacteristic
392C2×C4#95311yescharacteristic
393C8—42no—
394C8—122no—
395C8—241no—
396C2×C3—22no—
397C2×C3—82no—
398C2×C3—161no—
399C2×C3#201011yescharacteristic
400C2×C3#201011yescharacteristic
401C2×C3#201211yescharacteristic
402C22—21no—
403C22—21no—
404C22—44no—
405C22—42no—
406C22—82no—
407C22—81no—
408C22—81no—
409C22—81no—
410C22—81no—
411C22—81no—
412C22#515812yes—
413C22#522311yescharacteristic
414C4—21no—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—81no—
419C4—81no—
420C4—122no—
421C4—242no—
422C4—241no—
423C4#515312yes—
424C4#515411yescharacteristic
425C4#515511yescharacteristic
426C3#1001311yescharacteristic, complemented, Sylow
427C2—22no—
428C2—82no—
429C2—161no—
430C2#7171511yescharacteristic
431C2#7204511yescharacteristic
432C2#7211811yescharacteristic
subgroupG/H
centerC2×C4#940
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#3100C2
socleC22×C3#392
hypercenter#3100D3
hypocenterC3#10013
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3100

Complements

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

NH ≅ G/Ntype
2. #83315C2semidirect
3. #82348C2semidirect
4. #71816C2semidirect
5. #71598C2semidirect
6. #70770C2semidirect
7. #67737C2semidirect
426. C3#10013semidirect

Maximal subgroups

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

subgroupindex
C3×#31002
#833152
#823482
#718162
#715982
#707702
#677372
#100133