Finite Group Databasebeta

← #10697580

Subgroups — #10697580

Order 768.

578 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 103 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×#3226C211yesmaximal, characteristic
2#83314C211yesmaximal, characteristic
3#83239C211yesmaximal, characteristic, complemented
4#71811C211yesmaximal, characteristic
5#71773C211yesmaximal, characteristic, complemented
6#70766C211yesmaximal, characteristic, complemented
7#67746C211yesmaximal, characteristic, complemented
8#10243—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×#951—21no—
12C2×#944—21no—
13C2×C4×Dic6C2211yescharacteristic
14C3×#500—21no—
15C3×#486—21no—
16C3×#480—22no—
17C3×#474—22no—
18C3×#431C2211yescharacteristic
19#5232—21no—
20#5229—21no—
21#5090—21no—
22#5075—21no—
23#5054—22no—
24#4983—22no—
25#4972—22no—
26C4×C4⋊Dic3C2211yescharacteristic
27#4925C2211yescharacteristic
28#4924C2211yescharacteristic
29#4547—22no—
30#4537—22no—
31#4484—22no—
32#3226D311yescharacteristic
33#3037—31no—
34#2546—31no—
35#1546—31no—
36#1502—31no—
37#1479—31no—
38#1303—31no—
39C2×C3×#41—21no—
40C22×C3×Q8—21no—
41C22×Dic6D411yescharacteristic
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8D411yescharacteristic
47C3×#127D411yescharacteristic
48C3×#124—21no—
49C3×#121—21no—
50C3×#121—21no—
51C3×C4×D4—21no—
52C3×C4×D4—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4D412yes—
55C2×C3×C42C2311yescharacteristic
56#951—42no—
57#951—42no—
58#944—42no—
59#944—42no—
60C2×#216—22no—
61C2×#214—22no—
62C2×C4⋊Dic3—21no—
63C2×C4⋊Dic3D411yescharacteristic
64C2×Dic3⋊C4—24no—
65C2×Dic3⋊C4—21no—
66C2×C4×Dic3—21no—
67C2×C4×Dic3—21no—
68C2×#208—21no—
69C4×Dic6—21no—
70C4×Dic6—21no—
71C4×Dic6—21no—
72C4×Dic6—21no—
73C3×#107—24no—
74C3×C4⋊C8D412yes—
75C3×#102—24no—
76C3×C22⋊C8—21no—
77C3×C4×C8—21no—
78#835—22no—
79#832—22no—
80C3⋊#95—24no—
81C3⋊#95D412yes—
82#830—21no—
83C4⋊(C3⋊C8)—21no—
84C4⋊(C3⋊C8)—21no—
85C4⋊(C3⋊C8)D412yes—
86C4×C3⋊C8—21no—
87C2×#137—61no—
88#532C2×D311yescharacteristic
89C2×C4⋊Q8—61no—
90C2×C4⋊Q8C2×D311yescharacteristic
91C2×C4×Q8—31no—
92#500—21no—
93#486—21no—
94#480—22no—
95#474—22no—
96#464—62no—
97#462—61no—
98#460—61no—
99#431—31no—
100#431C2×D311yescharacteristic
101#423—31no—
102#417—61no—
103#398—62no—
104#391—61no—
105#389—62no—
106C4×C4⋊C4—31no—
107#332—62no—
108#327—62no—
109Q8⋊C8—62no—
110C3×#41—41no—
111C3×#41—41no—
112C3×#41—41no—
113C3×#41—41no—
114C2×C3×Q8—21no—
115C2×C3×Q8—41no—
116C2×C3×Q8—41no—
117C2×C3×Q8—41no—
118C2×C3×Q8—41no—
119C2×C3×Q8—41no—
120C2×C3×Q8—41no—
121C2×C3×D4—21no—
122C2×C3×D4—21no—
123C2×C3×D4—21no—
124C2×C3×D4—41no—
125C22×C3×C4—22no—
126C22×C3×C4—41no—
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×Dic3—22no—
131C22×Dic3—21no—
132C22×Dic3—21no—
133C2×Dic6—21no—
134C2×Dic6—21no—
135C2×Dic6—21no—
136C2×Dic6—21no—
137C2×Dic6—21no—
138C2×Dic6D812yes—
139C2×C3×C8—22no—
140C3×C4⋊C4—22no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—22no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—41no—
145C3×C4⋊C4—41no—
146C3×C22⋊C4—41no—
147C3×C22⋊C4—41no—
148C3×C42C2×D411yescharacteristic
149C3×C42C2×D411yescharacteristic
150C3×C42C2×D411yescharacteristic
151C3×C42#4111yescharacteristic
152#216—44no—
153#214—44no—
154C4⋊Dic3—24no—
155C4⋊Dic3—21no—
156C4⋊Dic3—21no—
157Dic3⋊C4—48no—
158Dic3⋊C4—41no—
159Dic3⋊C4—41no—
160C4×Dic3—41no—
161C4×Dic3—41no—
162C4×Dic3—41no—
163C4×Dic3—41no—
164#208—41no—
165#208—41no—
166C2×C3⋊C8—22no—
167C2×C3⋊C8—41no—
168C2×#41—21no—
169C22×Q8—21no—
170C22×Q8—31no—
171#137—122no—
172#137—122no—
173C2×Q16—62no—
174C2×SD16—62no—
175C2×M16—61no—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—122no—
181C4⋊Q8—122no—
182C4⋊Q8#6711yescharacteristic
183#127#6711yescharacteristic
184#124—21no—
185#121—21no—
186#121—21no—
187C4×Q8—61no—
188C4×Q8—61no—
189C4×Q8—61no—
190C4×Q8—61no—
191C4×D4—21no—
192C4×D4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—31no—
195C2×C4⋊C4—61no—
196C2×C4⋊C4—64no—
197C2×C4⋊C4—61no—
198C2×C4⋊C4#6712yes—
199C2×C42—61no—
200C2×C42—61no—
201C2×C42C22×D311yescharacteristic
202#107—24no—
203C4⋊C8—32no—
204C4⋊C8—61no—
205C4⋊C8—61no—
206C4⋊C8#6712yes—
207#103—62no—
208#102—24no—
209#102—62no—
210C22⋊C8—21no—
211C22⋊C8—61no—
212C4×C8—21no—
213C4×C8—61no—
214#95—32no—
215#95—64no—
216C23×C3—41no—
217C23×C3#12011yescharacteristic
218C3×Q8—41no—
219C3×Q8—41no—
220C3×Q8—81no—
221C3×Q8—81no—
222C3×Q8—81no—
223C3×Q8—81no—
224C3×D4—41no—
225C3×D4—41no—
226C3×D4—42no—
227C3×D4—81no—
228C3×D4—81no—
229C3×D4—81no—
230C2×C3×C4—22no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—42no—
234C2×C3×C4—42no—
235C2×C3×C4—42no—
236C2×C3×C4—81no—
237C2×C3×C4—81no—
238C2×C3×C4#12011yescharacteristic
239C2×C3×C4#12011yescharacteristic
240C2×C3×C4#12111yescharacteristic
241C2×C3×C4#12111yescharacteristic
242C2×C3×C4#12111yescharacteristic
243C2×C3×C4#12711yescharacteristic
244C2×C3×C4C2×D811yescharacteristic
245C2×C3×C4#13511yescharacteristic
246C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
247C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
248C2×Dic3—24no—
249C2×Dic3—22no—
250C2×Dic3—42no—
251C2×Dic3—41no—
252C2×Dic3—42no—
253C2×Dic3—41no—
254C2×Dic3—42no—
255C2×Dic3—41no—
256Dic6—22no—
257Dic6—22no—
258Dic6—41no—
259Dic6—41no—
260C3×C8—42no—
261C3⋊C8—42no—
262C3⋊C8—81no—
263#41—41no—
264#41—41no—
265#41—41no—
266#41—41no—
267C2×Q8—21no—
268C2×Q8—32no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—61no—
276C2×Q8—61no—
277C2×Q8—61no—
278C2×Q8—61no—
279C2×Q8—61no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—41no—
284C22×C4—22no—
285C22×C4—41no—
286C22×C4—62no—
287C22×C4—61no—
288C22×C4—61no—
289C22×C4D3×D411yescharacteristic
290C22×C4D3×D411yescharacteristic
291C22×C4C2×#6711yescharacteristic
292Q16—124no—
293SD16—124no—
294M16—121no—
295M16—121no—
296C2×C8—22no—
297C2×C8—62no—
298C2×C8—121no—
299C4⋊C4—22no—
300C4⋊C4—21no—
301C4⋊C4—22no—
302C4⋊C4—21no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—64no—
306C4⋊C4—61no—
307C4⋊C4—61no—
308C4⋊C4—128no—
309C4⋊C4—121no—
310C4⋊C4—121no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C42—121no—
314C42—121no—
315C42—121no—
316C42—121no—
317C42D3×D411yescharacteristic
318C42#23711yescharacteristic
319C42C2×#6711yescharacteristic
320C42C2×#6711yescharacteristic
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—81no—
324C22×C3—81no—
325C22×C3—81no—
326C22×C3#39211yescharacteristic
327C22×C3#44711yescharacteristic
328C22×C3#44911yescharacteristic
329C3×C4—21no—
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—82no—
333C3×C4—81no—
334C3×C4#44711yescharacteristic
335C3×C4#44911yescharacteristic
336C3×C4#45612yes—
337C3×C4#46611yescharacteristic
338C3×C4#46911yescharacteristic
339Dic3—44no—
340Dic3—42no—
341Dic3—81no—
342C23—41no—
343C23#93711yescharacteristic
344Q8—41no—
345Q8—41no—
346Q8—62no—
347Q8—62no—
348Q8—81no—
349Q8—81no—
350Q8—81no—
351Q8—81no—
352Q8—121no—
353Q8—121no—
354D4—41no—
355D4—41no—
356D4—42no—
357D4—81no—
358D4—81no—
359D4—81no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—64no—
367C2×C4—62no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—122no—
371C2×C4—122no—
372C2×C4—121no—
373C2×C4—121no—
374C2×C4—122no—
375C2×C4—121no—
376C2×C4D3×D811yescharacteristic
377C2×C4#91111yescharacteristic
378C2×C4#91111yescharacteristic
379C2×C4#91211yescharacteristic
380C2×C4#93711yescharacteristic
381C2×C4#93811yescharacteristic
382C2×C4#93911yescharacteristic
383C2×C4#93911yescharacteristic
384C2×C4#94011yescharacteristic
385C2×C4#95311yescharacteristic
386C8—42no—
387C8—122no—
388C8—241no—
389C2×C3—21no—
390C2×C3—21no—
391C2×C3—161no—
392C2×C3#199911yescharacteristic
393C2×C3#200511yescharacteristic
394C2×C3#200911yescharacteristic
395C22—21no—
396C22—21no—
397C22—81no—
398C22—81no—
399C22—81no—
400C22#515611yescharacteristic
401C22#515811yescharacteristic
402C22#522311yescharacteristic
403C4—21no—
404C4—21no—
405C4—21no—
406C4—82no—
407C4—81no—
408C4—124no—
409C4—122no—
410C4—241no—
411C4#515611yescharacteristic
412C4#515711yescharacteristic
413C4#515811yescharacteristic
414C4#515911yescharacteristic
415C4#524612yes—
416C3#1024311yescharacteristic, complemented, Sylow
417C2—21no—
418C2—21no—
419C2—161no—
420C2#7186811yescharacteristic
421C2#7204311yescharacteristic
422C2#7211111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3226C2
socleC22×C3#392
prime radicalC3×#532C22
hypercenter#3226D3
hypocenterC3#10243
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3226

Complements

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

NH ≅ G/Ntype
3. #83239C2semidirect
5. #71773C2semidirect
6. #70766C2semidirect
7. #67746C2semidirect
416. C3#10243semidirect

Maximal subgroups

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

subgroupindex
C3×#32262
#833142
#832392
#718112
#717732
#707662
#677462
#102433